Answer:
[tex]\displaystyle \cos x \left(\csc x + \tan x\right) = \cot x + \sin x[/tex]
Step-by-step explanation:
We want to simplify:
[tex]\displaystyle \cos x \left(\csc x + \tan x\right)[/tex]
Recall that csc x = 1 / sin x and tan x = sin x / cos x. Substitute:
[tex]\displaystyle = \cos x \left(\frac{1}{\sin x} + \frac{\sin x }{\cos x}\right)[/tex]
Distribute:
[tex]\displaystyle = \frac{\cos x}{\sin x} + \frac{\cos x \sin x}{\cos x} \\ \\ \\ = \frac{\cos x}{\sin x} + \sin x[/tex]
Recall that cot x = cos x / sin x. Hence:
[tex]\displaystyle = \cot x + \sin x[/tex]
In conclusion:
[tex]\displaystyle \cos x \left(\csc x + \tan x\right) = \cot x + \sin x[/tex]
