Trigonometrical Ratios and Identities 1 Question 8
8. The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as
(a) $\sin A \cos A+1$
(b) $\sec A \operatorname{cosec} A+1$
(c) $\tan A+\cot A$
(d) $\sec A+\operatorname{cosec} A$
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Answer:
Correct Answer: 8. (d)
Solution:
- Given expression is
$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$
$$ \begin{aligned} & =\frac{\sin A}{\cos A} \times \frac{\sin A}{\sin A-\cos A}+\frac{\cos A}{\sin A} \times \frac{\cos A}{\cos A-\sin A} \\ & =\frac{1}{\sin A-\cos A} \frac{\sin ^{3} A-\cos ^{3} A}{\cos A \sin A} \\ & =\frac{\sin ^{2} A+\sin A \cos A+\cos ^{2} A}{\sin A \cos A} \\ & =\frac{1+\sin A \cos A}{\sin A \cos A}=1+\sec A \operatorname{cosec} A \end{aligned} $$
