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Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(V)
(i) \( (\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta} \)
(ii) \( \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A \)
(iii) \( \frac{\tan \theta}{\mid-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta \)
[Hint : Write the expression in terms of \( \sin \theta \) and \( \cos \theta \) ]
(iv) \( \frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A} \quad \) [Hint : Simplify LHS and RHS separately]
(v) \( \frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\operatorname{cosec} A+\cot A \), using the identity \( \operatorname{cosec}^{2} A=1+\cot ^{2} A \).
(vi) \( \sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A \)
(vii) \( \frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta \)
(viii) \( (\sin \mathrm{A}+\operatorname{cosec} \mathrm{A})^{2}+(\cos \mathrm{A}+\sec \mathrm{A})^{2}=7+\tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A} \)
(ix) \( (\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A} \)
[Hint : Simplify LHS and RHS separately]
\( P^{1} \)
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