# Cot2θ+tanθ = ? Write Answer With Explanation...

May 2, 2018

$= \csc 2 \theta$

#### Explanation:

We know that,

$\left(1\right) \tan 2 x = \frac{2 \tan x}{1 - {\tan}^{2} x}$

$\left(2\right) \sin 2 x = \frac{2 \tan x}{1 + {\tan}^{2} x}$

Using $\left(1\right) \mathmr{and} \left(2\right)$

$\cot 2 \theta + \tan \theta = \frac{1}{\tan} \left(2 \theta\right) + \tan \theta$

$= \frac{1}{\frac{2 \tan \theta}{1 - {\tan}^{2} \theta}} + \tan \theta$

$= \frac{1 - {\tan}^{2} \theta}{2 \tan \theta} + \tan \theta$

$= \frac{1 - {\tan}^{2} \theta + 2 {\tan}^{2} \theta}{2 \tan \theta}$

$= \frac{1 + {\tan}^{2} \theta}{2 \tan \theta}$

$= \frac{1}{\frac{2 \tan \theta}{1 + {\tan}^{2} \theta}}$

$= \frac{1}{\sin 2 \theta}$

$= \csc 2 \theta$

May 2, 2018

$= \csc 2 \theta$

#### Explanation:

We know that,

color(red)((1)cosAcosB+sinAsinB=cos(A-B)

color(blue)((2)cscA=1/sinA

Now,

$\cot 2 \theta + \tan \theta = \frac{\cos 2 \theta}{\sin 2 \theta} + \sin \frac{\theta}{\cos} \theta$

=color(red)((cos2thetacostheta+sin2thetasintheta))/(sin2thetacostheta)..tocolor(red)(Apply(1)

$= \frac{\textcolor{red}{\left(\cos \left(2 \theta - \theta\right)\right)}}{\sin 2 \theta \cos \theta}$

$= \cos \frac{\theta}{\sin 2 \theta \cos \theta}$

=color(blue)(1/(sin2theta)...toApply(2)

=color(blue)(csc2theta