If $3\cos \theta  + 4\sin \theta  = 5$ then $3\sin \theta  - 4\cos \theta $ is

If $x = \cos 10^\circ \cos 20^\circ \cos 40^\circ ,$ then the value of $x$ is

$A, B, C$ are the angles of a triangle, then ${\sin ^2}A + {\sin ^2}B + {\sin ^2}C - 2\cos A\,\cos B\,\cos C = $

In the figure, $\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(B E)=4(A B)$. If the area of $\triangle CAB$ is $2 \sqrt{3}-3$ unit $^2$, when $\frac{\theta_2}{\theta_1}$ is the largest, then the perimeter (in unit) of $\triangle CED$ is equal to $...........$.

Prove that $\frac{\cos 7 x+\cos 5 x}{\sin 7 x-\sin 5 x}=\cot x$

$2{\cos ^2}\theta - 2{\sin ^2}\theta = 1$, then $\theta  =$ .......$^o$