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Question

The expression $$ \displaystyle \frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$$ can be written as:


A
secAcosecA+1
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B
tanA+cotA
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C
secA+cosecA
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D
sinAcosA+1
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Solution

The correct option is B $$\sec A \text{cosec} A + 1$$
$$\displaystyle\frac{tan A}{1-cot A} + \displaystyle\frac{cot A}{1-tan A} $$
$$=\displaystyle \frac{1}{cot A(1-cot A)}-\frac{cot^{2}A}{1-\cot A}$$
$$=\displaystyle \frac{1-\cot^{3}A}{\cot A(1-\cot A)}$$
$$= \displaystyle \frac{\cos ec^{2}A+\cot A}{\cot A}$$
$$=1+sec A cosecA$$

Mathematics

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