Question

How do you verify `(sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x`?

Answer 1

Pretty sure the question is `(sinx)(tanxcosx-cotxcos x)=1-2cos^2x` ,or else it will be not provable.

Explanation:

Some basic knowledge to begin with:
1. `sin^2x+cos^2x=1`
2. `tanx=sinx/cosx`
3. `cotx=cosx/sinx`

Let's start from the left hand side
`(sinx)(tanxcosx-cotxcos x)`

`=sinxtanxcosx-sinxcotxcosx`

`=sinx(sinx/cosx)cosx-sinx(cosx/sinx)cosx`

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

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

`=1-2cos^2x`

Answer 2

`"see explanation"`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)tanx=sinx/cosx" and "cotx=cosx/sinx`

`•color(white)(x)sin^2x+cos^2x=1`

`rArrsin^2x=1-cos^2x`

`"consider the left side"`

`"distributing the parenthesis"`

`sinx xxsinx/cancel(cosx)xxcancel(cosx)-cancel(sinx)xxcosx/cancel(sinx)xxcosx`

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

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

`=1-2cos^2x=" right side "rArr"verified"`