Trigonometric Ratio and Identites

Let $\tan A, \tan B$, where $A, B \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2-2 x-5=0$. Then $20 \sin ^2\left(\frac{A+B}{2}\right)$ is equal to:

Let $\overrightarrow{a_k}=\left(\tan \theta_k\right) \hat{i}+\hat{j}$ and $\overrightarrow{b_k}=\hat{i}-\left(\cot \theta_k\right) \hat{j}$, where $\theta_k=\frac{2^{k-1} \pi}{2^n+1}$, for some $n \in \mathbb{N}, n>5$. Then the value of $\f...

If $\mathrm{A}=\frac{\sin 3^{\circ}}{\cos 9^{\circ}}+\frac{\sin 9^{\circ}}{\cos 27^{\circ}}+\frac{\sin 27^{\circ}}{\cos 81^{\circ}}$ and $\mathrm{B}=\tan 81^{\circ}-\tan 3^{\circ}$, then $\frac{\mathrm{B}}{\mathrm{A}}$ is equal to
$\_\_\_\_...

If $\sin\left(\frac{\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \sin\left(\frac{7\pi}{18}\right) = K$, then the value of $\sin\left(\frac{10K\pi}{3}\right)$ is:

Let $P = \{ \theta \in [0, 4\pi] : \tan^2 \theta \neq 1 \}$ and $S = \{ a \in \mathbb{Z} : 2(\cos^8 \theta - \sin^8 \theta) \sec 2 \theta = a^2, \theta \in P \}$. Then $n(S)$ is:

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{\sin ^2 \mathrm{C}}{\sin ^2 \mathrm{~A}}=1, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \in\left(0, \frac{\pi}{2}\right)$, then

If $\cot x=\frac{5}{12}$ for some $x \in\left(\pi, \frac{3 \pi}{2}\right)$, then $\sin 7 x\left(\cos \frac{13 x}{2}+\sin \frac{13 x}{2}\right)+\cos 7 x\left(\cos \frac{13 x}{2}-\sin \frac{13 x}{2}\right)$ is equal to

The value of $\frac{\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}}$ is equal to

Let $\frac{\pi}{2}<\theta<\pi$ and $\cot \theta=-\frac{1}{2 \sqrt{2}}$. Then the value of
\(\sin \left(\frac{15 \theta}{2}\right)(\cos 8 \theta+\sin 8 \theta)+\cos \left(\frac{15 \theta}{2}\right)(\cos 8 \theta-\sin 8 \theta)\)
is equal t...

$$ \text { If } \frac{\cos ^2 48^{\circ}-\sin ^2 12^{\circ}}{\sin ^2 24^{\circ}-\sin ^2 6^{\circ}}=\frac{\alpha+\beta \sqrt{5}}{2} \text {, where } \alpha, \beta \in \mathbb{N} \text {, then } \alpha+\beta \text { is equal to ___________} $...

Let $\cos (\alpha+\beta)=-\frac{1}{10}$ and $\sin (\alpha-\beta)=\frac{3}{8}$, where $0<\alpha<\frac{\pi}{3}$ and $0<\beta<\frac{\pi}{4}$. If $\tan 2 \alpha=\frac{3(1-r \sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r, s \in N$, then $r+s$ is equal to ...

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to :

If for $\theta \in\left[-\frac{\pi}{3}, 0\right]$, the points $(x, y)=\left(3 \tan \left(\theta+\frac{\pi}{3}\right), 2 \tan \left(\theta+\frac{\pi}{6}\right)\right)$ lie on $x y+\alpha x+\beta y+\gamma=0$, then $\alpha^2+\beta^2+\gamma^2$ ...

If $10 \sin ^4 \theta+15 \cos ^4 \theta=6$, then the value of $\frac{27 \operatorname{cosec}^6 \theta+8 \sec ^6 \theta}{16 \sec ^8 \theta}$ is

If $\sin x + \sin^2 x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + \tan^{12} x) + 3(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to:

If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, ...

If \(\sin x=-\frac{3}{5}\), where \(\pi< x <\frac{3 \pi}{2}\), then \(80\left(\tan ^2 x-\cos x\right)\) is equal to

If the value of \(\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}\) is \(\frac{a \sqrt{5}-b}{c}\), where \(a, b, c\) are natural numbers and \(\operatorname{gcd}(a, c)=1\), then \(a+b+c\) is equal to :

Suppose \(\theta \in\left[0, \frac{\pi}{4}\right]\) is a solution of \(4 \cos \theta-3 \sin \theta=1\). Then \(\cos \theta\) is equal to :

The number of solutions, of the equation \(e^{\sin x}-2 e^{-\sin x}=2\), is :

For \(\alpha, \beta \in(0, \pi / 2)\), let \(3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)\) and a real number \(k\) be such that \(\tan \alpha=k \tan \beta\). Then, the value of \(k\) is equal to

Let the set of all $a \in \mathbf{R}$ 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 pqr is equal to ____________.

If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan \mathrm{C}=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<\mathrm{A}, \mathrm{B}, \mathrm{C}<\frac{\pi}{2}$, then $\mathrm{...

The value of \(36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)\) is :

The value of \(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}\) is __________.

If \(\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a\), then the value of \(\left( {a + {1 \over a}} \right)\) is :

Let \(f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )\) and $$S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}...

The set of all values of \(\lambda\) for which the equation \({\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda\) has a real solution \(x\), is :

\(96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}\) is equal to :

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

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

\(\alpha = \sin 36^\circ\) is a root of which of the following equation?

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

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

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

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

If \(\sin \theta + \cos \theta = {1 \over 2}\), then 16(sin(2\(\theta\)) + cos(4\(\theta\)) + sin(6\(\theta\))) is equal to :

If \(\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)\) are in arithmetic progression and \(\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)\) are also in arithmetic progress...

The number of integral values of 'k' for which the equation \(3\sin x + 4\cos x = k + 1\) has a solution, k\(\in\)R is ___________.

The value of $$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} ...

The value of \(\cot {\pi \over {24}}\) is :

If 0 < x, y < \(\pi\) and cosx + cosy \(-\) cos(x + y) = \({3 \over 2}\), then sinx + cosy is equal to :

If
\({e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}\)
satisfies the equation t2 - 9t + 8 = 0, then the value of
$${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)...

If 15sin4\(\alpha\) + 10cos4\(\alpha\) = 6, for some \(\alpha\)\(\in\)R, then the value of 27sec6\(\alpha\) + 8cosec6\(\alpha\) is equal to :

If for x \(\in\) \(\left( {0,{\pi \over 2}} \right)\), log10sinx + log10cosx = \(-\)1 and log10(sinx + cosx) = \({1 \over 2}\)(log10 n \(-\) 1), n > 0, then the value of n is equal to :

The value of
\({\cos ^3}\left( {{\pi \over 8}} \right)\)\({\cos}\left( {{3\pi \over 8}} \right)\)+\({\sin ^3}\left( {{\pi \over 8}} \right)\)\({\sin}\left( {{3\pi \over 8}} \right)\)
is :

If \(x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta }\) and \(y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta }\) for
0 < \(\theta\) < \({\pi \over 4}\), then :

If \({{\sqrt 2 \sin \alpha } \over {\sqrt {1 + \cos 2\alpha } }} = {1 \over 7}\) and \(\sqrt {{{1 - \cos 2\beta } \over 2}} = {1 \over {\sqrt {10} }}\)
\(\alpha ,\beta \in \left( {0,{\pi \over 2}} \right)\) then tan(\(\alpha\) + 2$$\bet...