Question

Cot(x) + tan(x) = sec(x) csc(x) for all real numbers x ≠ ???? 2 n for integer n.

Answer 1

To prove: $\cot x+\tan x=\sec x\csc x$

where $x\neq \dfrac{\pi}{2}(2n-1)$

Using trigonometry formula:

$\cot x=\dfrac{\cos x}{\sin x}$

$\tan x=\dfrac{\sin x}{\cos x}$

$\sec x=\dfrac{1}{\cos x}$

$\csc x=\dfrac{1}{\sin x}$

Taking Left hand side

$\Rightarrow \dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}$

$\Rightarrow \dfrac{\cos^2 x+\sin^2x}{\sin x\cos x}$

$\Rightarrow \dfrac{1}{\sin x\cos x}$         $\because \cos^2 x+\sin^2x=1$

$\Rightarrow \sec x\csc x$  

Hence proved