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Prove the following identity, where the...

Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
(vii) `(sintheta-2sin^3theta)/(2cos^3theta-costheta)=tantheta`

Text Solution

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`L.H.S. =(sintheta - 2sin^3theta)/(2cos^3theta-costheta)`
`=(sintheta(1-2sin^2theta))/(costheta(2cos^2theta-1))`
`=tantheta(cos^2theta+sin^2theta-2sin^2theta)/(2cos^2theta-cos^2theta-sin^2theta)`
`=tantheta(cos^2theta-sin^2theta)/(cos^2theta-sin^2theta)`
`=tantheta=R.H.S.`
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