Trigonometric Functions

Simplify the following expression:

$\left({\sin }^2\left(1°\right)+{\sin }^2\left(3°\right)+{\sin }^2\left(5°\right)+...........+{\sin }^2\left(85°\right)+{\sin }^2\left(87°\right)+{\sin }^2\left(89°\right)\right)$

If $0\le \theta \le 2\mathrm{\pi }$, solve the equation:

$4\tan \left(\theta \right)+2=2\tan \left(\theta \right)$

${\sec }^2\theta =\frac{4xy}{{(x+y)}^2}$ is true if and only if

Given  $A={\sin }^2\theta +{\cos }^4\theta$ , then for all real values of  $\theta$

Let  $\theta \in \left(0,\frac{\pi }{4}\right)$  and  $t_1={\left(\tan \theta \right)}^{\tan \theta },t_2={\left(\tan \theta \right)}^{\cot \theta },t_3={\left(\cot \theta \right)}^{\tan \theta }\mathrm{and}{t}_4={\left(\cot \theta \right)}^{\cot \theta }$ , then –

Given both $\theta$ and  $\varphi$  are acute angles  $\sin \theta =\frac{1}{2}, \cos \varphi =\frac{1}{3}$, then the value of  $(\theta +\varphi )$ belongs to 

If  $\alpha +\beta +\gamma =\mathrm{}\mathrm{}\mathrm{}2\pi ,\mathrm{then}$ –

${\sec }^{-1}\left[\frac{{\mathrm{x}}^2+1}{{\mathrm{x}}^2-1}\right]=?$

If ${\sin }^2\theta =K^2$ $\left(0<k<1\right)$ and $180°<\theta <270°$ then $\frac{k}{\sqrt{1-k^2}}\cot \theta +\sqrt{1-k^2}.\mathrm{sec\theta }=$

The trigonometric function $y=\tan x$ in the $\mathrm{II}$ quadrant

The value of ${\sin }^{2}51°+{\sin }^{2}39°$ is 

FInd the value of ${\sin }^2\left(3\mathrm{\pi }\right)+{\cos }^2\left(4\mathrm{\pi }\right)+{\tan }^2\left(5\mathrm{\pi }\right).$

The value of $\sum _{K=1}^7{\tan }^2\left(\frac{K\mathrm{\pi }}{16}\right)$ is

If $ABCD$ is a quadrilateral then ${\sec }^2\left(\frac{A+B}{4}\right)-{\cot }^2\left(\frac{C+D}{4}\right)=?$

Find the value of $\frac{\sin 135°-\cos 120°}{\sin 135°+\cos 120°}$.

If $\sec \theta +\tan \theta =4,$ then $\cos \theta =$

The value of $\cos \left(-{870}^0\right)$ is

The maximum value of $1+\sin \left(\frac{\mathrm{\pi }}{4}+\theta \right)+2\cos \left(\frac{\mathrm{\pi }}{4}-\theta \right), \theta \in R$ equals