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8. Introduction to Trigonometry
hard
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta$
Option A
Option B
Option C
Option D
Solution
$\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos \theta+\cos \theta}=\tan \theta$
$L.H.S.=\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}$
$=\frac{\sin \theta\left(1-2 \sin ^{2} \theta\right)}{\cos \theta\left(2 \cos ^{2} \theta-1\right)}$
$=\frac{\sin \theta \times\left(1-2 \sin ^{2} \theta\right)}{\cos \theta \times\left\{2\left(1-\sin ^{2} \theta\right)-1\right\}}$
$=\frac{\sin \theta \times\left(1-2 \sin ^{2} \theta\right)}{\cos \theta \times\left(1-2 \sin ^{2} \theta\right)}$
$=\tan \theta= R \cdot H.S.$
Standard 10
Mathematics
