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\( 16\sin(20^\circ)\sin(40^\circ)\sin(80^\circ) \) is equal to:

Let a vertical tower \( AB \) of height \( 2h \) stands on a horizontal ground. Let from a point \( P \) on the ground a man can see upto height \( h \) of the tower with an angle of elevation \( 2\alpha \). When from \( P \), he moves a distance \( d \) in the direction of \( \overrightarrow{AP} \), he can see the top \( B \) of the tower with an angle of elevation \( \alpha \). If \( d = \sqrt{7}\,h \), then \( \tan \alpha \) is equal to:

Let \( S = \left\{ \theta \in [-\pi, \pi] - \left\{ \pm \frac{\pi}{2} \right\} : \sin \theta \tan \theta + \tan \theta = \sin 2\theta \right\} \). If \( T = \ \displaystyle \sum_{\theta \in S} \cos 2\theta \), then \( T + n(S) \) is equal to:

\( \text{The value of } \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \text{ is equal to:} \)

From the base of a pole of height \( 20 \, \text{meter} \), the angle of elevation of the top of a tower is \( 60^\circ \). The pole subtends an angle \( 30^\circ \) at the top of the tower. Then the height of the tower is:

Let \( S = \left\{ \theta \in \left(0, \dfrac{\pi}{2}\right) : \displaystyle\sum_{m=1}^{9} \sec\left(\theta + (m-1)\frac{\pi}{6}\right)\sec\left(\theta + \frac{m\pi}{6}\right) = -\frac{8}{\sqrt{3}} \right\} \). Then

\( \displaystyle 2 \sin\!\left(\frac{\pi}{22}\right)\sin\!\left(\frac{3\pi}{22}\right)\sin\!\left(\frac{5\pi}{22}\right)\sin\!\left(\frac{7\pi}{22}\right)\sin\!\left(\frac{9\pi}{22}\right) \) 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 \( AB \) and \( PQ \) be two vertical poles, \( 160 \, \text{m} \) apart from each other. Let \( C \) be the middle point of \( B \) and \( Q \), which are feet of these two poles. Let \( \frac{\pi}{8} \) and \( \theta \) be the angles of elevation from \( C \) to \( P \) and \( A \), respectively. If the height of pole \( PQ \) is twice the height of pole \( AB \), then \( \tan^{2}\theta \) is equal to:

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 ________.

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JEE Main 2022 25th June Evening Shift

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