Question

Prove the trigonometric identity? `tanx(1-cot^2x)+cotx(1-tan^2x)=0`

Answer 1

Prove trig identity.

Explanation:

Replace in the equation `tan x = (sin x)/(cos x)`, and `cot x = (cos x)/(sin x)`
`(sin x)/(cos x)(1 - (cos^2 x)/(sin^2 x)) + (cos x)/(sin x) (1 - sin^2 x/(cos^2 x)) =`
`(sin x)/(cos x) - (cos x)/(sin x) + (cos x)/(sin x) - (sin x)/(cos x) = 0` OK

Answer 2

Applying identity `tanxcotx=1`

`LHS=tanx(1-cot^2x)+cotx(1-tan^2x)`

`=tanx-tanxcot^2x+cotx-cotxtan^2x`

`=tanx-1*cotx+cotx-1*tanx`

`=0=RHS`

Proved