# help with proving trig indentity

• Apr 23rd 2014, 09:57 AM
cricket71
help with proving trig indentity
I need to prove that cosx+cotx/secx+tanx = cosxcotx, any help would be greatly appreciated!
• Apr 23rd 2014, 10:27 AM
romsek
Re: help with proving trig indentity
$\cos(x)+\dfrac{\cot(x)}{\sec(x)}+\tan(x) \overset{?}{=} \cos(x)\cot(x)$

$\cos(x)+\dfrac{\dfrac{\cos(x)}{\sin(x)}}{\dfrac{1 }{\cos(x)}}+\dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \cos(x)\dfrac{\cos(x)}{\sin(x)}$

$\cos(x)+\dfrac{\cos^2(x)}{\sin(x)} + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \dfrac{\cos^2(x)}{\sin(x)}$

$\cos(x) + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} 0$

$\cos^2(x) \overset{?}{=} -\sin(x)$

No. These two expressions are not equal.

If you meant

$\cos(x)+\dfrac{\cot(x)}{\sec(x)+\tan(x)} \overset{?}{=} \cos(x)\cot(x)$

Then I suggest you learn to use parentheses and use the method I followed above.
• Apr 23rd 2014, 01:08 PM
Soroban
Re: help with proving trig indentity
Hello, cricket71!

Quote:

$\text{Prove: }\:\frac{\cos x+\cot x}{\sec x+\tan x} \:=\: \cos x\cot x$

We have: . $\frac{\cos x+\cot x}{\sec x+\tan x}$

Multiply by $\tfrac{\sec x - \tan x}{\sec x - \tan x}$

. . $\frac{\cos x+\cot x}{\sec x+\tan x} \cdot \frac{\sec x - \tan x}{\sec x - \tan x}$

. . $=\;\frac{\overbrace{\cos x\sec x}^{1} - \overbrace{\cos x\tan x}^{\sin x} +\overbrace{\cot x\sec x}^{\frac{1}{\sin x}} - \overbrace{\cot x\tan x}^1}{\underbrace{\sec^2\!x - \tan^2\!x}_1}$

. . $=\;\frac{1}{\sin x} - \sin x \;=\;\frac{1-\sin^2\!x}{\sin x} \;=\;\frac{\cos^2\!x}{\sin x}$

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