Question 1

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
A)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Question 2

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$

Accepted Answer

A)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
A)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Question 3

Simplify the expression algebraically.​ $$3 \sin \left( \frac { \pi } { 2 } - x \right)$$ ​

Accepted Answer

A)  $\frac { 1 } { 3 } \cos x$ 
B)  $3 \cos x$ 
C)  $- \frac { 1 } { 3 } \cos x$ 
D)  $- 3 \cos x$ 
E)  $3 \sin x$ 
A)  $\frac { 1 } { 3 } \cos x$ 
B)  $3 \cos x$ 
C)  $- \frac { 1 } { 3 } \cos x$ 
D)  $- 3 \cos x$ 
E)  $3 \sin x$

Question 4

Verify the given identity. $\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$

Accepted Answer

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Question 5

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$

Accepted Answer

A)  $\sin \left( - 110 ^ { \circ } \right)$ 
B)  $\sin \left( - 10 ^ { \circ } \right)$ 
C)  $\sin \left( 100 ^ { \circ } \right)$ 
D)  $\sin \left( 45 ^ { \circ } \right)$ 
E)  $\sin \left( 55 ^ { \circ } \right)$ 
A)  $\sin \left( - 110 ^ { \circ } \right)$ 
B)  $\sin \left( - 10 ^ { \circ } \right)$ 
C)  $\sin \left( 100 ^ { \circ } \right)$ 
D)  $\sin \left( 45 ^ { \circ } \right)$ 
E)  $\sin \left( 55 ^ { \circ } \right)$

Question 6

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )

Accepted Answer

A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E)  $\cos ( u + v ) = \frac { 56 } { 65 }$ 
A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E)  $\cos ( u + v ) = \frac { 56 } { 65 }$

Question 7

Find the exact value of the given expression. $$\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { \sqrt { 3 } } { 2 }$

Question 8

Simplify the expression algebraically.​ $$\frac { 9 } { \sqrt { 2 } } \cos \left( \frac { 5 \pi } { 4 } - x \right)$$ ​

Accepted Answer

A) - $\frac { 9 } { 2 }$ $( \cos x + \sin x )$ 
B)  $\frac { 9 } { 2 }$ $( \cos x - \sin x )$ 
C)  $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } + \sin \frac { 5 x } { 4 } \right)$ 
D)  $\frac { 9 } { 2 }$ $( \sin x - \cos x )$ 
E)  $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } - \sin \frac { 5 x } { 4 } \right)$ 
A) - $\frac { 9 } { 2 }$ $( \cos x + \sin x )$ 
B)  $\frac { 9 } { 2 }$ $( \cos x - \sin x )$ 
C)  $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } + \sin \frac { 5 x } { 4 } \right)$ 
D)  $\frac { 9 } { 2 }$ $( \sin x - \cos x )$ 
E)  $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } - \sin \frac { 5 x } { 4 } \right)$

Question 9

Find the exact solutions of the given equation in the interval [0,2π). sin 2x = sin x
​

Accepted Answer

A)  $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
E)  $x = 0$ 
A)  $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
E)  $x = 0$

Question 10

Simplify the expression algebraically.​ $$6 \sin \left( \frac { \pi } { 2 } + x \right)$$ ​

Accepted Answer

A)  $6 \cos x$ 
B)  $- 6 \cos x$ 
C)  $6 \sin x$ 
D)  $- \frac { 1 } { 6 } \cos x$ 
E)  $\frac { 1 } { 6 } \cos x$ 
A)  $6 \cos x$ 
B)  $- 6 \cos x$ 
C)  $6 \sin x$ 
D)  $- \frac { 1 } { 6 } \cos x$ 
E)  $\frac { 1 } { 6 } \cos x$

Question 11

Simplify the following expression algebraically.​ $$\cos \left( \frac { 3 \pi } { 2 } - x \right)$$ ​

Accepted Answer

A)  $\sin x$ 
B)  $\frac { 3 } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $- \sin x$ 
E)  $- \frac { 3 } { 2 } \sin x$ 
A)  $\sin x$ 
B)  $\frac { 3 } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $- \sin x$ 
E)  $- \frac { 3 } { 2 } \sin x$

Question 12

Simplify the given expression algebraically.​ $\cos ( x - \pi )$

Accepted Answer

A)  $\sin x$ 
B)  $\cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$ 
E) 1 
A)  $\sin x$ 
B)  $\cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$ 
E) 1

Question 13

Write the given expression as the cosine of an angle.​ $$\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$$

Accepted Answer

A)  $\cos \left( 45 ^ { \circ } \right)$ 
B)  $\cos \left( 5 ^ { \circ } \right)$ 
C)  $\cos \left( - 80 ^ { \circ } \right)$ 
D)  $\cos \left( 40 ^ { \circ } \right)$ 
E)  $\cos \left( 85 ^ { \circ } \right)$ 
A)  $\cos \left( 45 ^ { \circ } \right)$ 
B)  $\cos \left( 5 ^ { \circ } \right)$ 
C)  $\cos \left( - 80 ^ { \circ } \right)$ 
D)  $\cos \left( 40 ^ { \circ } \right)$ 
E)  $\cos \left( 85 ^ { \circ } \right)$

Question 14

Find the exact value of the given expression. $$\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)$$

Accepted Answer

A)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
B)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
A)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
B)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 15

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 3 , b = 7 , B = 2$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

Accepted Answer

A)  $\sqrt { 58 }$ $\cos ( 2 \theta + 0.4049 )$ 
B)  $\sqrt { 58 }$ $\cos ( 2 \theta - 0.4049 )$ 
C) 7 $\cos ( 2 \theta - 0.4049 )$ 
D) 3 $\cos ( 2 \theta - 0.4049 )$ 
E) 3 $\cos ( 2 \theta + 0.4049 )$ 
A)  $\sqrt { 58 }$ $\cos ( 2 \theta + 0.4049 )$ 
B)  $\sqrt { 58 }$ $\cos ( 2 \theta - 0.4049 )$ 
C) 7 $\cos ( 2 \theta - 0.4049 )$ 
D) 3 $\cos ( 2 \theta - 0.4049 )$ 
E) 3 $\cos ( 2 \theta + 0.4049 )$

Question 16

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 9 , b = 2 , B = 2$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

Accepted Answer

A)  $\sqrt { 85 }$ $\cos ( 3 \theta - 1.3521 )$ 
B) 9 $\cos ( 3 \theta + 1.3521 )$ 
C) 9 $\cos ( 3 \theta - 1.3521 )$ 
D)  $\sqrt { 85 }$ $\cos ( 3 \theta + 1.3521 )$ 
E) 2 $\cos ( 3 \theta - 1.3521 )$ 
A)  $\sqrt { 85 }$ $\cos ( 3 \theta - 1.3521 )$ 
B) 9 $\cos ( 3 \theta + 1.3521 )$ 
C) 9 $\cos ( 3 \theta - 1.3521 )$ 
D)  $\sqrt { 85 }$ $\cos ( 3 \theta + 1.3521 )$ 
E) 2 $\cos ( 3 \theta - 1.3521 )$

Question 17

Use a graphing utility to select correct graph of $y _ { 1 }$ and $y _ { 2 }$ in the same viewing window.Use the graphs to determine whether $y _ { 1 } = y _ { 2 }$ .Explain your reasoning.​ $$y _ { 1 } = \sin ( x + 6 ) , y _ { 2 } = \sin ( x ) + \sin ( 6 )$$ ​

Accepted Answer

A) ​   No, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different. 
B) ​   Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different. 
C) ​   Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are same. 
D) ​   No, $$y _ { 1 } \neq y _ { 2 }$$ because their graphs are Same. 
E) ​   No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different. 
A) ​   No, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different. 
B) ​   Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different. 
C) ​   Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are same. 
D) ​   No, $$y _ { 1 } \neq y _ { 2 }$$ because their graphs are Same. 
E) ​   No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different.

Question 18

Simplify the expression algebraically.​ $$\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$$ ​

Accepted Answer

A)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$ 
B)  $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$ 
C)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$ 
D)  $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$ 
E)  $\cos ^ { 2 } x - \sin ^ { 2 } y$ 
A)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$ 
B)  $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$ 
C)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$ 
D)  $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$ 
E)  $\cos ^ { 2 } x - \sin ^ { 2 } y$

Question 19

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$ ​

Accepted Answer

A)  $\cos \left( 25 ^ { \circ } \right)$ 
B)  $\cos \left( 85 ^ { \circ } \right)$ 
C)  $\cos \left( 35 ^ { \circ } \right)$ 
D)  $\cos \left( 60 ^ { \circ } \right)$ 
E)  $\cos \left( - 50 ^ { \circ } \right)$ 
A)  $\cos \left( 25 ^ { \circ } \right)$ 
B)  $\cos \left( 85 ^ { \circ } \right)$ 
C)  $\cos \left( 35 ^ { \circ } \right)$ 
D)  $\cos \left( 60 ^ { \circ } \right)$ 
E)  $\cos \left( - 50 ^ { \circ } \right)$

Question 20

Find the expression as the tangent of an angle.​ $$\frac { \tan 130 ^ { \circ } - \tan 30 ^ { \circ } } { 1 + \tan 130 ^ { \circ } \tan 30 ^ { \circ } }$$ ​

Accepted Answer

A)  $\tan ^ { - 1 } 100 ^ { \circ }$ 
B)  $\tan ^ { - 1 } 130 ^ { \circ }$ 
C)  $\tan 160 ^ { \circ }$ 
D)  $\tan 100 ^ { \circ }$ 
E)  $\tan 30 ^ { \circ }$ 
A)  $\tan ^ { - 1 } 100 ^ { \circ }$ 
B)  $\tan ^ { - 1 } 130 ^ { \circ }$ 
C)  $\tan 160 ^ { \circ }$ 
D)  $\tan 100 ^ { \circ }$ 
E)  $\tan 30 ^ { \circ }$

Exam 34: Sum and Difference Formulas

Find the exact value of the given expression using a sum or difference formula. sin⁡285∘\sin 285 ^ { \circ }sin285∘

Find the exact value of the given expression. sin⁡(2π3−5π4)\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)sin(32π​−45π​)

Simplify the expression algebraically.​ 3sin⁡(π2−x)3 \sin \left( \frac { \pi } { 2 } - x \right)3sin(2π​−x) ​

Verify the given identity. cos⁡(x+y)cos⁡(x−y)=cos⁡2x−sin⁡2y\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } ycos(x+y)cos(x−y)=cos2x−sin2y

Write the given expression as the sine of an angle. sin⁡45∘cos⁡55∘−sin⁡55∘45∘\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }sin45∘cos55∘−sin55∘45∘

Find the exact value of cos⁡(u+v)\cos ( u + v )cos(u+v) given that sin⁡u=513\sin u = \frac { 5 } { 13 }sinu=135​ and cos⁡v=−45\cos v = - \frac { 4 } { 5 }cosv=−54​ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. sin⁡105∘cos⁡345∘−sin⁡345∘cos⁡105∘\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }sin105∘cos345∘−sin345∘cos105∘

Simplify the expression algebraically.​ 92cos⁡(5π4−x)\frac { 9 } { \sqrt { 2 } } \cos \left( \frac { 5 \pi } { 4 } - x \right)2​9​cos(45π​−x) ​

Find the exact solutions of the given equation in the interval [0,2π). sin 2x = sin x ​

Simplify the expression algebraically.​ 6sin⁡(π2+x)6 \sin \left( \frac { \pi } { 2 } + x \right)6sin(2π​+x) ​

Simplify the following expression algebraically.​ cos⁡(3π2−x)\cos \left( \frac { 3 \pi } { 2 } - x \right)cos(23π​−x) ​

Simplify the given expression algebraically.​ cos⁡(x−π)\cos ( x - \pi )cos(x−π)

Write the given expression as the cosine of an angle.​ cos⁡45∘cos⁡40∘+sin⁡45∘sin⁡40∘\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }cos45∘cos40∘+sin45∘sin40∘

Find the exact value of the given expression. sin⁡(π3−π4)\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)sin(3π​−4π​)

Simplify the expression algebraically.​ cos⁡(7x+4y)cos⁡(7x−4y)\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )cos(7x+4y)cos(7x−4y) ​

Write the given expression as the cosine of an angle. cos⁡60∘cos⁡25∘−sin⁡60∘sin⁡25∘\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }cos60∘cos25∘−sin60∘sin25∘ ​

Find the expression as the tangent of an angle.​ tan⁡130∘−tan⁡30∘1+tan⁡130∘tan⁡30∘\frac { \tan 130 ^ { \circ } - \tan 30 ^ { \circ } } { 1 + \tan 130 ^ { \circ } \tan 30 ^ { \circ } }1+tan130∘tan30∘tan130∘−tan30∘​ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$

Simplify the expression algebraically. $3 \sin \left( \frac { \pi } { 2 } - x \right)$

Verify the given identity. $\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. $\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$

Simplify the expression algebraically. $\frac { 9 } { \sqrt { 2 } } \cos \left( \frac { 5 \pi } { 4 } - x \right)$

Find the exact solutions of the given equation in the interval [0,2π). sin 2x = sin x

Simplify the expression algebraically. $6 \sin \left( \frac { \pi } { 2 } + x \right)$

Simplify the following expression algebraically. $\cos \left( \frac { 3 \pi } { 2 } - x \right)$

Simplify the given expression algebraically. $\cos ( x - \pi )$

Write the given expression as the cosine of an angle. $\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$

Find the exact value of the given expression. $\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)$

Simplify the expression algebraically. $\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$

Find the expression as the tangent of an angle. $\frac { \tan 130 ^ { \circ } - \tan 30 ^ { \circ } } { 1 + \tan 130 ^ { \circ } \tan 30 ^ { \circ } }$