Question

tan/1-tan -cot /1-cot= cos+sin/cos-sin

Answer 1

Answer:

Step-by-step explanation:

We have to prove $\frac{(tan\theta)}{(1-tan\theta)}-\frac{cot\theta}{1-cot\theta}=\frac{(cos\theta+sin\theta)}{(cos\theta-sin\theta)}$

We consider the left hand side of the equation

$\frac{(tan\theta)}{(1-tan\theta)}-\frac{cot\theta}{1-cot\theta}=\frac{\frac{sin\theta}{cos\theta}}{(1-\frac{sin\theta}{cos\theta})}-\frac{\frac{cos\theta}{sin\theta}}{(1-\frac{cos\theta}{sin\theta})}$

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

                          = $\frac{sin\theta}{cos\theta-sin\theta}-\frac{cos\theta}{[-(cos\theta-sin\theta)]}$

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

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

                          = Right hand side of the equation

Hence proved.

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