Question

tan theta upon 1 - cos theta + cot theta upon 1 - tan theta

Answer 1

here is your answer
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Answer 2

Answer:

Given Expression  $=\:/frac{tan\,\theta}{1-cot\,\theta}+\frac{cot\,\theta}{1-tan\,\theta}$    

To  simplify the expression

Consider,

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

$\implies\:\frac{\frac{sin\theta}{cos\theta}}{1-\frac{cos\theta}{sin\theta}}+\frac{\frac{cos\theta}{sin\theta}}{1-\frac{sin\theta}{cos\theta}}$  

$\implies\:\frac{\frac{sin\theta}{cos\theta}}{\frac{sin\theta-cos\theta}{sin\theta}}+\frac{\frac{cos\theta}{sin\theta}}{\frac{cos\theta-sin\theta}{cos\theta}}$

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

$\implies\:\frac{sin^2\theta}{cos\theta(sin\theta-cos\theta)}}-\frac{cos^2\theta}{sin\theta(sin\theta-cos\theta)}}$

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

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

$\implies\:\frac{sin^2\theta+cos^2\theta+sin\theta\,cos\theta}{sin\theta\,cos\theta}}$

$\implies\:\frac{sin^2\theta}{sin\theta\,cos\theta}+\frac{cos^2\theta}{sin\theta\,cos\theta}+\frac{sin\theta\,cos\theta}{sin\theta\,cos\theta}}$

$\implies\:\frac{sin\theta}{cos\theta}+\frac{cos\theta}{sin\theta}+1$

$\implies\:\tan\theta+cot\theta+1$