Exam 7: Analytic Trigonometry

Correct Answer:

Verified

$\cos ^ { 2 } 2 \theta - \sin ^ { 2 } 2 \theta = \cos 4 \theta$

Correct Answer:

Verified

$\frac { 1 - \cos x } { \sin x } + \frac { \sin x } { 1 - \cos x } = \frac { 2 } { \sin x } = 2 \csc x$

Correct Answer:

Verified

$\cos ^ { 2 } \pi x - \sin ^ { 2 } \pi x = 0$ $\Leftrightarrow$ $2 \cos ^ { 2 } \pi x - 1 = 0$ $\Leftrightarrow$ $\cos ^ { 2 } \pi x = \frac { 1 } { 2 }$ $\Leftrightarrow$ $\cos \pi x = \pm \frac { \sqrt { 2 } } { 2 }$ , which has general solutions $\pi x = \frac { \pi } { 4 } + 2 k \pi$ , $\pi x = \frac { 7 \pi } { 4 } + 2 k \pi$ , $\pi x = \frac { 3 \pi } { 4 } + 2 k \pi$ , and $\pi x = \frac { 5 \pi } { 4 } + 2 k \pi$ , these can be written $x = \frac { 1 } { 4 } + 2 k$ , $x = \frac { 7 } { 4 } + 2 k$ , $x = \frac { 3 } { 4 } + 2 k$ , $x = \frac { 5 } { 4 } + 2 k$ , or briefly as $x = \frac { 1 } { 4 } + \frac { k } { 2 }$ , where $k$ is an integer.