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