Trigonometric Identities

Find the exact value of ${\tan }^2\frac{\pi }{16}+{\tan }^2\frac{3\pi }{16}+{\tan }^2\frac{5\pi }{16}+{\tan }^2\frac{7\pi }{16}$.

In triangle $ABC, \cos A+2\cos B+\cos C=2$, then-

If $\left(3-4{\sin }^{2}1\right)\left(3-4{\sin }^{2}3\right)\left(3-4{\sin }^2{3}^2\right)\dots \dots \left(3-4{\sin }^2\left(3^{n-1}\right)\right)=\sin a/\sin b$, where $n\in N \text{and} a, b$ are integers in radian, then the digit at the unit place of $(a+b)$ may be-

If $2\tan 10°+\tan 50°=2x, \tan 20°+\tan 50°=2y, 2\tan 10°+\tan 70°=2w$ and $\tan 20°+\tan 70°=2z$, then which of the following is/are true -

If $\tan \frac{\theta }{2}=cosec\theta -\sin \theta$, then -

If  $\sin \theta +\sqrt{\sin \theta +\sqrt{\sin \theta +\sqrt{\sin \theta +\dots \dots \infty }}}={\sec }^4\alpha$, then $\sin \theta$ is equal to -

Let $A,B,C$ are $3$ angles such that $\cos A+\cos B+\cos C=0$ and if $\cos A\cos B\cos C=\lambda (\cos 3A+\cos 3B+\cos 3C)$, then $\lambda$ is equal to -

If $\sin x+\sin y=a \& \cos x+\cos y=b$, then which of the following may be true.

The value of $\cos \frac{\mathrm{\pi }}{19}+\cos \frac{3\mathrm{\pi }}{19}+\cos \frac{5\mathrm{\pi }}{19}+\dots \dots +\cos \frac{17\mathrm{\pi }}{19}$ is equal to

$\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$ is equal to

If $\frac{\cos \alpha }{\cos A}+\frac{\sin \alpha }{\sin A}=\frac{\cos \beta }{\cos A}+\frac{\sin \beta }{\sin A}=1$, where $\alpha$ and $\beta$ do not differ by an even multiple of $\mathrm{\pi }$, then $\frac{\cos \alpha \cos \beta }{{\cos }^{2}A}+\frac{\sin \alpha \sin \beta }{{\sin }^{2}A}=$_____

If $A,B,C,D$ are the angles of a quadrilateral, then $\frac{\tan A+\tan B+\tan C+\tan D}{\cot A+\cot B+\cot C+\cot D}=$

If $\alpha =\frac{2\pi }{7},$ then $\tan \alpha ·\tan 2\alpha +t\mathrm{an}2\alpha ·\tan 4\alpha +\tan 4\alpha ·\tan \alpha =$

Let $f\left(x\right)={\sin }^2(x+\alpha )+{\sin }^2(x+\beta )-2\cos \left(\alpha -\beta \right)\sin \left(x+\alpha \right)\sin \left(x+\beta \right).$ Which of the following is TRUE?

 If $\alpha , \beta , \gamma$ are acute angles and $\cos \theta =\frac{ \sin \beta }{\sin \alpha }, \cos \varphi =\frac{ \sin \gamma }{\sin \alpha }$and $\cos (\theta -\varphi )=\sin \beta \sin \gamma ,$ then the value of${\tan }^2\alpha - {\tan }^2\beta - {\tan }^2\gamma$ is