Question

tan theta upon 1 minus cot theta + cot theta upon 1 - tan theta

Answer 1

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Answer 2

Answer:

Value of given Expression is $cosec\,\theta\:sec\,\theta}+1$

Step-by-step explanation:

Given Expression:

$\frac{tan\,\theta}{1-cot\,\theta}+\frac{cot\,\theta}{1-tan\,\theta}$

To find: Value of the given expression.

Consider,

$\frac{tan\,\theta}{1-cot\,\theta}+\frac{cot\,\theta}{1-tan\,\theta}$

$=\frac{\frac{sin\,\theta}{cos\,\theta}}{1-\frac{cos\,\theta}{sin\,\theta}}+\frac{\frac{cos\,\theta}{sin\,\theta}}{1-\frac{sin\,\theta}{cos\,\theta}}$

$=\frac{\frac{sin\,\theta}{cos\,\theta}}{\frac{sin\,\theta-cos\,\theta}{sin\,\theta}}+\frac{\frac{cos\,\theta}{sin\,\theta}}{\frac{cos\,\theta-sin\,\theta}{cos\,\theta}}$

$=\frac{sin^2\,\theta}{cos\,\theta(sin\,\theta-cos\,\theta)}+\frac{cos^2\,\theta}{sin\,\theta(cos\,\theta-sin\,\theta)}$

$=\frac{sin^3\,\theta-cos^3\,\theta}{sin\,\theta\:cos\,\theta(sin\,\theta-cos\,\theta)}$

$=\frac{(sin\,\theta-cos\,\theta)(sin^2\,\theta+cos^2\,\theta+sin\,\theta\:cos\,\theta)}{sin\,\theta\:cos\,\theta(sin\,\theta-cos\,\theta)}$

$=\frac{1+sin\,\theta\:cos\,\theta}{sin\,\theta\:cos\,\theta}$

$=\frac{1}{sin\,\theta\:cos\,\theta}+\frac{sin\,\theta\:cos\,\theta}{sin\,\theta\:cos\,\theta}$

$=cosec\,\theta\:sec\,\theta}+1$

Therefore, Value of given Expression is $cosec\,\theta\:sec\,\theta}+1$