Question

How do you prove `tan^2x(1-sin^2x)=(sin^2x-cos2x+cos^2x)/2`?

Answer 1

Use identities and simplify. Try simplifying Left Hand Side (LHS) and Right Hand Side separately if you are unable to progress. It might end up giving same result.

Explanation:

Before trying to prove this review the following identities.

1) `tan(theta) = sin(theta)/cos(theta)`
2) `cos^2(theta) + sin^2(theta) = 1`
3) `1-sin^2(theta) = cos^2(theta)`
4) `cos(2theta) = cos^2(theta) - sin^2(theta)`
5) `cos(2theta) = 2cos^2(theta) - 1`
6)`cos(2theta) = 1-2sin^2(theta)`
(7) `2sin^2(theta) = 1-cos(2theta))`

LHS = `tan^2(x)(1-sin^2(x))`
`=sin^2(x)/cos^2(x) (cos^2(x))`
`=sin^2(x)`

RHS `= (sin^2(x) -cos(2x) + cos^2(x))/2`

`=(sin^2(x)+cos^2(x)-cos(2x))/2`
`=(1-cos(2x))/2`
`=(2sin^2(x))/2`
`=sin^2(x)`

LHS = RHS thus proved.