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=(\sin x)/(\cos x)`

We can replace in the equation the tangent function:

`(1)/(\cos x)-\sin x\cdot(\sin x)/(\cos x)=\cos x`

Reordering the equation, we get:

`(1)/(\cos x)\cdot\lbrack1-\sin x\cdot\sin x\rbrack=\cos x`
`(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:

`(1)/(\cos x)\lbrack1-\sin ^2x\rbrack=\cos x`
`(1)/(\cos x)\cos ^2x=\cos x`

Finally, we get:

`\cos x=\cos x`