Question

prove that cos theta divided by 1 minus tan theta + sin theta divided by 1 minus cot theta is equals to sin theta + cos theta​

Answer 1

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Answer 2

$\dfrac{\cos\theta}{1-\tan\theta}+\dfrac{\sin\theta}{1-\cot\theta} =\sin\theta+\cos\theta$ Proved.

Step-by-step explanation:

Consider the provided expression.

$\dfrac{\cos\theta}{1-\tan\theta}+\dfrac{\sin\theta}{1-\cot\theta} =\sin\theta+\cos\theta$

Consider the LHS

$\dfrac{\cos\theta}{1-\tan\theta}+\dfrac{\sin\theta}{1-\cot\theta}$

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

$=\dfrac{\cos^2\theta}{\cos\theta-\sin\theta}+\dfrac{\sin^2\theta}{\sin\theta-\cos\theta}$

$=\dfrac{\cos^2\theta}{\cos\theta-\sin\theta}-\dfrac{\sin^2\theta}{\cos\theta-\sin\theta}$

$=\dfrac{\cos^2\theta-\sin^2\theta}{\cos\theta-\sin\theta}$

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

$=\cos\theta+\sin\theta$

LHS=RHS

Hence, proved

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