Trigonometric Identities

If $\cos x+{\cos }^{2}x=1$, then what is the value of ${\sin }^{4}x+{\sin }^{2}x$?

If $\theta$ is a positive acute angle and $\tan \left(2\theta \right)\tan \left(3\theta \right)=1$, then the value of the expression $\left(2{cos}^2\frac{5\theta }{2}-1\right)$.

What should come in the place of ($?$) for the following expression?

$\cot \left(10°\right)·\cot \left(20°\right)·\cot \left(60°\right)·\cot \left(70°\right)·\cot \left(80°\right)=?$

If ${\tan }^2\left(\theta \right) =1 – a^2$, then find the value of $\sec \left(\theta \right) + {\tan }^3\left(\theta \right)\mathrm{cosec}\left(\theta \right)$.

Find the least value of $(4{\sec }^2\theta +9{\mathrm{cosec}}^2\theta )$.

$\frac{\tan 3x}{\tan x}$never lies between

If $\sec \left(\theta \right)+\tan \left(\theta \right)=5$, then the value of $\frac{\tan \left(\theta \right)+1}{\tan \left(\theta \right)-1}$ is:

Find the value of $3\sin 15° - 4 {\sin }^3 15°$

$\cos 60° . \cos 30° + \sin 60° . \sin 30° =$_____.

What should come in place of ($?$) for the below expression?

$\frac{\sin \left(\theta \right)-\cos \left(\theta \right)+1 }{\sin \left(\theta \right)+\cos \left(\theta \right)-1} =?$

If $\frac{\cos \alpha }{\cos \beta }=\mathrm{m}$ and $\frac{\cos \alpha }{\sin \beta }=\mathrm{n}$, then value of $({\mathrm{m}}^2+{\mathrm{n}}^2) {\cos }^2\beta$_____.

If $\mathrm{A}+\mathrm{C}=\mathrm{B}$, then $\mathrm{tanA} \mathrm{tanB} \mathrm{tanC}$ is equal to -

If $\tan \left(2\theta \right)· \tan \left(3\theta \right) =1$, where $0°< \theta < 90°$, then find the value of $\theta$.

If $0º< \theta < 90º$, the value of $(\sin \theta + \cos \theta )$ is _____.

If $\tan 2\theta \tan 4\theta =1$ so find $\tan 3\theta$.

What should come in the place of ($?$) for the trigonometric expression?

$\sin 20°\times \sin 40°\times \sin 60°\times \sin 80° = ?$

If ${\tan }^2\left(\theta \right)=1-e^2$, then the value of $\sec \left(\theta \right)+{\tan }^3\left(\theta \right)\csc \left(\theta \right)$ is

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

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