Trigonometric Identities

Let  $\mathrm{f}\left(\mathrm{x}\right)={\sin }^6\mathrm{x}+{\cos }^6\mathrm{x}+\mathrm{k}({\sin }^4\mathrm{x}+{\cos }^4\mathrm{x})$  for some real number $\mathrm{k},$ then the value of $\mathrm{k}$ for which $\mathrm{f}\left(\mathrm{x}\right)$ is constant for all values of $\mathrm{x}$ is

If  $\alpha +\beta =\frac{\pi }{2}$ and  $\beta +\gamma =\alpha$ , then  $\tan \alpha$  equals –

The value of the expression  $\sqrt{3}\mathrm{cosec}20°-\sec 20°$ is equal to

$\left(1+\cos \frac{\pi }{8}\right)\left(1+\cos \frac{\mathrm{}3\pi }{8}\right)\left(1+\cos \frac{\mathrm{}5\pi }{8}\right)\left(1+\cos \frac{\mathrm{}7\pi }{8}\right)$ is equal to –

If $\alpha +\beta +\gamma =2\pi$, then

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

In the figure, ${\mathrm{\theta }}_1+{\mathrm{\theta }}_2=\frac{\mathrm{\pi }}{2}$ and $\sqrt{3}\left(\mathrm{BE}\right)=4\left(\mathrm{AB}\right)$. If the area of $∆\mathrm{CAB}$ is $2\sqrt{3}-3 {\mathrm{unit}}^2$, when $\frac{{\mathrm{\theta }}_2}{{\mathrm{\theta }}_1}$ is the largest, then the perimeter (in unit) of $∆\mathrm{CED}$ is equal to

The value of  $\tan 9^{{}^{\mathrm{o}}}-\tan {27}^{{}^{\mathrm{o}}}-\tan {63}^{{}^{\mathrm{o}}}+\tan {81}^{{}^{\mathrm{o}}}$ is _____.

The value of the expression $\frac{{\sin }^2\theta +{\cos }^4\theta }{{\sin }^4\theta +{\cos }^2\theta }$ is always

If $\tan y=\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}$, the find $\sin 2y$.

${\cos }^4\left(\frac{\mathrm{\pi }}{24}\right)-{\sin }^4\left(\frac{\mathrm{\pi }}{24}\right)$ is equal to

If ${\sin }^6\theta +{\cos }^6\theta =\sin 2\theta$ then find the value of $\cos 4\theta$

The value of ${\sin }^{2}70°-{\sin }^{2}50°$ is

$\frac{\cos 10°+\sin 10°}{\cos 10°-\sin 10°}=$

If $0<\alpha <30°$, and the value of $E={\log }_3{\left(\frac{8\cos \left(30°-2\alpha \right)-{\displaystyle \frac{8}{\cot \alpha +\cot \left(30°-\alpha \right)}}}{\sqrt{3}}\right)}^2$ lies between two consecutive integer $a$ and $b$, then find the value of $a+b$.

The value of ${\left(\cos {65}^{°}+\sqrt{3}\sin 5^{°}+\cos 5^{°}\right)}^2=\lambda {\cos }^2{25}^{°}$; then value of $\lambda$ be

If $\cos 4°\cos 8°\cos 12°\cos 16°....\cos 84°\cos 88°=2^{-n}$, then $n=\_\_\_\_\_\_\_$