The number of solutions of $\cos x = \sin x$, such that $- 4 \pi \leq x \leq 4 \pi$ is
(1) 4
(2) 6
(3) 8
(4) 12

Let $S = \left\{ \theta \in [ 0,2 \pi ] : 8 ^ { 2 \sin ^ { 2 } \theta } + 8 ^ { 2 \cos ^ { 2 } \theta } = 16 \right\}$. Then $n ( S ) + \sum _ { \theta \in \mathrm { S } } \left( \sec \left( \frac { \pi } { 4 } + 2 \theta \right) \operatorname { cosec } \left( \frac { \pi } { 4 } + 2 \theta \right) \right)$ is equal to:
(1) 0
(2) $- 2$
(3) $- 4$
(4) 12

Let $S = \left\{ \theta \in \left( 0 , \frac { \pi } { 2 } \right) : \sum _ { m = 1 } ^ { 9 } \sec \left( \theta + ( m - 1 ) \frac { \pi } { 6 } \right) \sec \left( \theta + \frac { m \pi } { 6 } \right) = - \frac { 8 } { \sqrt { 3 } } \right\}$. Then
(1) $\mathrm { S } = \left\{ \frac { \pi } { 12 } \right\}$
(2) $S = \left\{ \frac { 2 \pi } { 3 } \right\}$
(3) $\sum _ { \theta \in S } \theta = \frac { \pi } { 2 }$
(4) $\sum _ { \theta \in S } \theta = \frac { 3 \pi } { 4 }$

Let $S = \left[ - \pi , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 2 } , - \frac { \pi } { 4 } , - \frac { 3 \pi } { 4 } , \frac { \pi } { 4 } \right\}$. Then the number of elements in the set $A = \{ \theta \in S : \tan \theta ( 1 + \sqrt { 5 } \tan ( 2 \theta ) ) = \sqrt { 5 } - \tan ( 2 \theta ) \}$ is $\_\_\_\_$.

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\sum_{\theta \in S} \sin\left(\theta + \frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.

Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[ p , q ]$ and $r = \tan 9 ^ { \circ } - \tan 27 ^ { \circ } - \frac { 1 } { \cot 63 ^ { \circ } } + \tan 81 ^ { \circ }$, then $p q r$ is equal to $\_\_\_\_$.

If $\alpha , - \frac { \pi } { 2 } < \alpha < \frac { \pi } { 2 }$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$, then the value of $\tan \alpha$ is
(1) $\frac { 10 - \sqrt { 10 } } { 6 }$
(2) $\frac { 10 - \sqrt { 10 } } { 12 }$
(3) $\frac { \sqrt { 10 } - 10 } { 12 }$
(4) $\frac { \sqrt { 10 } - 10 } { 6 }$

If $2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $0 , \frac { \mathrm { n } \pi } { 2 }$, for the least value of $\mathrm { n } \in \mathrm { N }$ then $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } \frac { \mathrm { k } } { 2 ^ { \mathrm { k } } }$ is equal to :
(1) $\frac { 1 } { 2 ^ { 15 } } 2 ^ { 14 } - 14$
(2) $\frac { 1 } { 2 _ { 1 } ^ { 14 } } 2 ^ { 15 } - 15$
(3) $1 - \frac { 15 } { 2 ^ { 13 } }$
(4) $\frac { 1 } { 2 ^ { 13 } } 2 ^ { 14 } - 15$

The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$

The sum of all values of $\theta \in [ 0,2 \pi ]$ satisfying $2 \sin ^ { 2 } \theta = \cos 2 \theta$ and $2 \cos ^ { 2 } \theta = 3 \sin \theta$ is
(1) $4 \pi$
(2) $\frac { 5 \pi } { 6 }$
(3) $\pi$
(4) $\frac { \pi } { 2 }$

How many real numbers $x$ satisfy $\sin\left(x + \frac{\pi}{6}\right) = \sin x + \sin\frac{\pi}{6}$ and $0 \leq x < 2\pi$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 or more

On the coordinate plane, the graph of the function $y = \sin x$ is symmetric about $x = \frac{\pi}{2}$, as shown in the figure. Find the value of $\theta$ in the range $0 < \theta \leq \pi$ that satisfies $\sin \theta = \sin\left(\theta + \frac{\pi}{5}\right)$.
(1) $\frac{\pi}{5}$
(2) $\frac{2\pi}{5}$
(3) $\frac{3\pi}{5}$
(4) $\frac{4\pi}{5}$
(5) $\pi$

Given that $0 < \mathrm { x } < \pi$,
$$\sin ^ { 4 } x = \cos ^ { 4 } x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equality?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\pi$
E) $2 \pi$

Let $0 \leq x \leq \frac { 3 \pi } { 2 }$. Given that
$$| \sin x | = \cos \left( 50 ^ { \circ } \right)$$
what is the sum of the $x$ values that satisfy this equation?
A) $\frac { 13 \pi } { 18 }$
B) $\frac { 11 \pi } { 90 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { 31 \pi } { 18 }$
E) $\frac { 20 \pi } { 9 }$

Let $\pi < x < 2\pi$,
$$\frac{2\cos^{2} x + 2\sin x}{5\sin(2x)} = \tan x$$
What is the sum of the real numbers $x$ that satisfy this equation?
A) $2\pi$ B) $3\pi$ C) $4\pi$ D) $5\pi$ E) $6\pi$

Let $0 \leq x \leq \pi$ and
$$\sqrt{2}\sin(4x) - \cos(8x) = 1$$
What is the sum of the $x$ values satisfying this equality?
A) $\dfrac{\pi}{3}$ B) $\dfrac{3\pi}{4}$ C) $\pi$ D) $\dfrac{3\pi}{2}$ E) $2\pi$