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\( \displaystyle 2 \sin\!\left(\frac{\pi}{22}\right)\sin\!\left(\frac{3\pi}{22}\right)\sin\!\left(\frac{5\pi}{22}\right)\sin\!\left(\frac{7\pi}{22}\right)\sin\!\left(\frac{9\pi}{22}\right) \) is equal to:

The angle of elevation of the top of a tower from a point \( A \) due north of it is \( \alpha \) and from a point \( B \) at a distance of \( 9 \) units due west of \( A \) is \( \cos^{-1}\!\left(\dfrac{3}{\sqrt{13}}\right) \). If the distance of the point \( B \) from the tower is \( 15 \) units, then \( \cot \alpha \) is equal to:

The number of solutions of the equation \( \sin x = \cos^{2} x \) in the interval \( (0,10) \) is ______.

The number of solutions of \( |\cos x|=\sin x \), such that \( -4\pi \le x \le 4\pi \) is:

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

A horizontal park is in the shape of a triangle \( OAB \) with \( AB = 16 \). A vertical lamps post \( OP \) is erected at the point \( O \) such that \( \angle PAO = \angle PBO = 15^\circ \) and \( \angle PCO = 45^\circ \), where \( C \) is the midpoint of \( AB \). Then \( (OP)^2 \) is equal to:

The number of solutions of the equation \( \cos\!\left(x+\frac{\pi}{3}\right)\cos\!\left(\frac{\pi}{3}-x\right)=\frac{1}{4}\cos^{2} 2x \), \( x \in [-3\pi,\,3\pi] \) is:

Let \( S=\left\{ \theta \in [0,2\pi] : 8^{2\sin^2\theta}+8^{2\cos^2\theta}=16 \right\}. \) Then \( n(S)+\displaystyle\sum_{\theta \in S}\big(\sec \left(\frac{\pi}{4}+2\theta\right)\csc \left(\frac{\pi}{4}+2\theta\right)\big) \) is equal to:

The number of values of \( x \) in the interval \( \left(\frac{\pi}{4}, \frac{7\pi}{4}\right) \) for which \( 14cosec^{2}x-2\sin^{2}x=21-4\cos^{2}x \) holds, is ________.

If \( \sin^2(10^\circ)\sin(20^\circ)\sin(40^\circ)\sin(50^\circ)\sin(70^\circ) = \alpha - \frac{1}{16}\sin(10^\circ) \), then \( 16 + \alpha^{-1} \) is equal to ____________ .

The value of \( 2\sin(12^{\circ})-\sin(72^{\circ}) \) is:

If \( \cot \alpha = 1 \) and \( \sec \beta = -\tfrac{5}{3} \), where \( \pi < \alpha < \tfrac{3\pi}{2} \) and \( \tfrac{\pi}{2} < \beta < \pi \), then the value of \( \tan(\alpha + \beta) \) and the quadrant in which \( \alpha + \beta \) lies, respectively are:

The number of elements in the set $S = \left\{ \theta \in [-4\pi, 4\pi] : 3\cos^2 2\theta - 10\cos^2 \theta + 6\cos 2\theta - 10\cos^2 \theta + 5 = 0 \right\}$ is ________.