Question

Who do I prove that the RHS is equal to LHS? Sin2x (tan x + cot x) = 2

Answer 1

Develop the left side by using the trig identity:
`sin 2x -= 2sin x*cos x`

`L.S = 2sin xcos x (sin x/cos x+ cos x/sin x)`

`= 2sin xcos x* ((cos^2 x + sin^2 x))/(sin xcos x )`

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

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

`therefore` Proved.

Answer 2

See below.

Explanation:

Using identities:

`color(red)(tanx =sinx/cosx)`

`color(red)(cotx=cosx/sinx)`

`color(red)(sin(2x)=2sinxcosx`

`color(red)(sin^2x+cos^2x=1)`
.......................................................................................................................................

`sin(2x)(tanx+cotx)`

`sin(2x)(sinx/cosx+cosx/sinx)`

`sin(2x)((sin^2x+cos^2x)/(sinxcosx))`

`sin(2x)((1)/(sinxcosx))`

`2sinxcosx((1)/(sinxcosx))`

`(2sinxcosx)/(sinxcosx)`

`(2cancel(sinxcosx))/(cancel(sinxcosx))=2`