# Verify trigonometric identity

• Mar 26th 2013, 05:28 PM
Unreal
Verify trigonometric identity
Hi!

Is my solution for the problem correct?
If so, what's the 'right' way to write it up?

$\displaystyle \cos x(\cot x + \tan x) = \csc x$

$\displaystyle \cos x\left(\frac{\text{cos}x}{\text{sin}x}\, +\, \frac{\text{sin}x}{\text{cos}x}\right)$

$\frac{\text{cos}^2x}{\text{sin}x}\,+\,\frac{\text{ cos}x\text{sin}x}{\text{cos}x}$

$\frac{1-\text{sin}^2x}{\text{sin}x}\,+\,\text{sin}x$

$\frac{{1-\text{sin}^2x\,+\,\text{sin}^2x}}{{\text{sin}x}}$

$\frac1{\text{sin}x} \equiv \csc$
• Mar 26th 2013, 05:45 PM
Soroban
Re: Verify trigonometric identity
Hello, Unreal

Your work is correct . . . It was a bit hard to follow.

Quote:

$\text{Prove: }\:\cos x(\cot x + \tan x) \:=\: \csc x$
$\text{Left side: }\:\cos x\left(\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}\right) \;\;=\;\;\cos x\cdot\frac{\overbrace{\cos^2\!x+\sin^2\!x}^{\text {This is 1}}}{\sin x\cos x}$

. . . . . . . . . . $=\;\;\cos x\cdot\frac{1}{\sin x\cos x} \;\;=\;\;\frac{1}{\sin x} \;\;=\;\; \csc x$