Question

How do you simplify $\cot x+\tan x$?

Answer 1

Simplify using trigonometric laws:

The expression $\cot x+\tan x$ can be simplified using trigonometric laws as follows,

$\cot x+\tan x=\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}$ $\left[\because tan\theta =\frac{sin\theta }{cos\theta },\cot \theta =\frac{cos\theta }{sin\theta }\right]$

$\Rightarrow \cot x+\tan x=\frac{{\left(\cos x\right)}^2+{\left(\sin x\right)}^2}{\sin x·\cos x}$

$\Rightarrow \cot x+\tan x=\frac{{\cos }^{2}x+{\sin }^{2}x}{\sin x·\cos x}$

$\Rightarrow \cot x+\tan x=\frac{1}{\sin x·\cos x}$ $\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]$

$\Rightarrow \cot x+\tan x=\frac{1}{\sin x·\cos x}\times \frac{2}{2}$

$\Rightarrow \cot x+\tan x=\frac{2}{2\sin x·\cos x}$

$\Rightarrow \cot x+\tan x=\frac{2}{\sin 2x}$ $[\because 2sin\theta cos\theta =sin\left(2\theta \right)]$

$\Rightarrow \cot x+\tan x=2\mathrm{cosec}\left(2x\right)$ $[\because \frac{1}{\mathrm{sin\theta }}=\mathrm{cosec\theta }]$

Hence, the simplest form of $\cot x+\tan x$ obtained using trigonometric laws is $2\mathrm{cosec}\left(2x\right)$.