Question

How do you simplify (Cosx)/(Secx)+(Tanx) ?

Answer 1

is the question like this?
\[\frac{ \cos x }{\sec x+\tan x }\]

Answer 2

Yes Exactly.

Answer 3

\[\frac{ \cos x }{\sec x+\tan x }\times \frac{ \sec x-\tan x }{\sec x-\tan x }=\frac{\cos x \left( \sec x-\tan x \right) }{ \sec ^2 x -\tan ^2x }\]
\[\left[ \sec ^2x-\tan ^2x=1 \right]\]
simplify it.

Answer 4

Alternately:

\[\frac{ \cos x }{\sec x+\tan x }=
\frac{ \cos x }{\dfrac1{\cos x}+\dfrac{\sin x}{\cos x} }\\\qquad\qquad\qquad=\frac{\cos^2x}{1+\sin x}\]

then use \(\sin^2 x+ \cos^2 x =1\\
\qquad\qquad\cos^2 x=1-\sin^2 x\)

Answer 5

and \(1-z^2=(1-z)(1+z)\)