Question

4.

Here is the proof of (sin x - cos x)2 = sec2 x - tan2 x - 2sin x cos x.
What is the missing line

Answer 1

See Explanation Below

Explanation:

Given

`(sin x - cos x)^2 = sec^2x - tan^2x - 2sinx.cos x.`

Required

Prove

To start with; we open the bracket on the left hand side

`(sin x - cos x)^2 = (sin x - cos x)(sin x - cos x)`

`(sin x - cos x)^2 = (sin x )(sin x - cos x)- (cos x)(sin x - cos x)`

`(sin x - cos x)^2 = sin^2 x -sinx cos x - sin xcos x + cos^2 x`

`(sin x - cos x)^2 = sin^2 x -2sinx cos x + cos^2 x`

Reorder

`(sin x - cos x)^2 = sin^2 x + cos^2 x - 2sinx cos x`

From trigonometry;

`sin^2x + cos^2x = 1`

So;

`(sin x - cos x)^2 = sin^2 x + cos^2 x - 2sinx cos x`

becomes

`(sin x - cos x)^2 = 1 - 2sinx cos x`

Also from trigonometry;

`sec^2x - tan^2x = 1`

So,

`(sin x - cos x)^2 = 1 - 2sinx cos x`

becomes

`(sin x - cos x)^2 = sec^2x - tan^2x - 2sinx cos x`

Proved