Question

Prove that cos theta/sin 2 theta - cos 2 theta/2 sin theta is equal to sin theta

Answer 1

❏QuesTion :-

$\bf Prove \: that : \\ \implies \: \frac{ \cos \theta}{ \sin2 \theta} - \frac{ \cos2 \theta}{2 \sin \theta} = \sin \theta$

❏Used formulas:-

$\star \: \boxed{ \lll \sin2 \theta = 2 \sin \theta \cos \theta \lll} \\ \\ \star \boxed{ \lll \: 1 - \cos2 \theta = 2 { \sin}^{2 } \theta \: \lll }$

❏Proof :-

Taking Left hand side ,

$\leadsto \: \frac{ \cos \theta}{ \sin2 \theta} - \frac{ \cos2 \theta}{2 \sin \theta} \: \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \because \: \boxed{ \sin2 \theta = 2 \sin \theta \cos \theta }\: \\ \therefore \\ \leadsto \: \frac{ \cos \theta}{ 2\sin \theta \cos \theta} - \frac{ \cos2 \theta}{2 \sin \theta} \\ \\ \leadsto \: \frac{1}{2 \sin \theta} - \frac{ \cos 2 \theta}{2 \sin \theta} \\ \\ \leadsto \: \frac{1 - \cos2 \theta}{2 \sin \theta} \\ \: \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \because \boxed{ 2 { \sin}^{2} \theta = 1 - \cos2 \theta} \: \\ \therefore \\ \leadsto \: \frac{2 { \sin}^{2} \theta }{2 \sin \theta} \\ \\ \leadsto \: \sin \theta \: \\ \\ \: \: \: \: \: \boxed{\frac{ \cos \theta}{ \sin2 \theta} - \frac{ \cos2 \theta}{2 \sin \theta} = \sin \theta \: } \\ \\ \mathcal{L.H.S = R.H.S \: } \: \\ \sf{Hence \: proved . \: }$