Question

verify the identity sec x- cos x = sin x tan x

Answer 1

1) Redefine secx using Reciprocal Identity
\[1/\cos x-\cos x=\sin x \tan x\]

2) LCD (Least Common Denominator)
\[(1-\cos ^2x)/\cos x=\sin x \tan x\]

3) Use the Basic Identity sin^2x + cos^2x = 1
\[\sin ^2x/\cos x\]

4) Factor out
\[(\sin x)(\sin x/\cos x)\]

5) Use the identity of tangent
\[(sinx)(\tan x)\]

VERIFIED. :)
You can also try to manipulate using the right hand side of the equation.
When proving, NEVER use both sides.
One at a time.
(Though there are cases that you'll prove using both sides)

Answer 2

\[ \sec x- \cos x = \frac{1}{\cos x} - \cos x = \frac{\sin^2 x }{\cos x} =\sin x \times \tan x \space \space [QED]\]

Answer 3

you guys are great thanks!