Question

Please Help!

Rewrite each expression in terms of sinθ, and simplify.

6.
`(tan(x))/(cot(x))`

7.
`cos(x)cot(x)+sin(x)`

Also, please show your work if you can!

Thanks in advance.

Answer 1

Explanation:

`6.\\\\(\tan x)/(\cot x)\qquad\text{use}\ \cot x=(1)/(\tan x)\\\\=(\tan x)/((1)/(\tan x))=\tan x\cdot(\tan x)/(1)=\tan x\cdot\tan x=\left(\tan x\right)^2\qquad\text{use}\ \tan x=(\sin x)/(\cos x)\\\\=\left((\sin x)/(\cos x)\right)^2\qquad\text{use}\ \left((a)/(b)\right)^n=(a^n)/(b^n)\\\\=(\sin^2x)/(\cos^2x)\qquad\text{use}\ \sin^2x+\cos^2x=1\to\cos^2x=1-\sin^2x\\\\=(\sin^2x)/(1-\sin^2x)`

`7.\\\\\cos x\cdot\cot x+\sin x\qquad\text{use}\ \cot x=(\cos x)/(\sin x)\\\\=\cos x\cdot(\cos x)/(\sin x)+\sin x=(\cos^2x)/(\sin x)+(\sin^2x)/(\sin x)=(\cos^2x+\sin^2x)/(\sin x)=(1)/(\sin x)`