Verify the identity by transforming the left-hand side into the right-hand side.  1-2sin^2(`theta/2)=2cos^2(theta/2)-1``` ``

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durbanville eNotes educator| Certified Educator

We need to make the LHS=RHS from `1-2sin^2(theta/2)=2cos^2(theta/2)-1`

LHS: `1-2sin^2(theta/2)`

We know that `sin^2 theta+cos^2 theta=1`

`therefore sin^2 theta = 1-cos ^2theta`

Replacing `theta` with `theta/2` in this instance in order to maintain the identity and in terms of the question:

`therefore sin^2(theta/2)=1-cos^2(theta/2)`

Now substitute into the original LHS:

`1-2 sin^2 (theta/2)`

`therefore = 1-2(1-cos^2 (theta/2))`

Expand and simplify taking care with the negative symbols:

`therefore= 1-2+2cos^2 (theta/2)`

`= -1+2cos^2 (theta/2)`


`therefore = 2cos^2 (theta/2)-1`

`therefore` LHS =RHS