Question

Prove that:
Cotx-tanx/cot²x-tan²x =1/2sin2x

Answer 1

Proof:

L.H.S:

$\frac{cot(x)-tan(x)}{cot^{2}(x)-tan^{2}(x)  }$

$=\frac{(cot(x)-tan(x))}{(cot(x)+tan(x))(cot(x)-tan(x))}$

$=\frac{1}{cot(x)+tan(x)}$

Expressing cot(x) as cos(x)/sin(x) and tan(x) as sin(x)/cos(x), we get

$=\frac{sin(x)cos(x)}{cos^{2}(x)+sin^{2}(x)}  = sin(x)cos(x)$

$=\frac{1}{2} (2sin(x)cos(x)) = \frac{1}{2} sin(2x)$

=R.H.S

Hence proved.