Question

I need help underrating how to prove identity and give the rule to use. I need help understand it better.

Answer 1

$\sec x-\sin x\tan x=\cos x$

0. Using the ,tangent identity,:

$\tan x=\frac{\sin x}{\cos x}$

We can replace in the equation the tangent function:

$\frac{1}{\cos x}-\sin x\cdot\frac{\sin x}{\cos x}=\cos x$

Reordering the equation, we get:

$\frac{1}{\cos x}\cdot\lbrack1-\sin x\cdot\sin x\rbrack=\cos x$$\frac{1}{\cos x}\lbrack1-\sin ^2x\rbrack=\cos x$

2. Using the Pythagorean identity:

$\sin ^2x+\cos ^2x=1$

...which can be reordered as:

$\cos ^2x=1-\sin ^2x^{}$

Replacing this in the latter expression we had:

$\frac{1}{\cos x}\lbrack1-\sin ^2x\rbrack=\cos x$$\frac{1}{\cos x}\cos ^2x=\cos x$

Finally, we get:

$\cos x=\cos x$