Question

WE HAVE TO PROVE LHS = RHS. PROVING STEP BY STEP WILL BE APPRECIATED

Answer 1

Answer:

your answer is as follows

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Step-by-step explanation:

$to \: prove \: that : \\ \frac{sin {}^{2}x }{1 - cot \: x} + \frac{cos {}^{2}x }{1 - tan \: x} = 1 + sin \: x.cos \: x \\ \\ LHS = \frac{sin {}^{2} x}{1 - cot \: x} + \frac{cos {}^{2} x}{1 - tan \: x} \\ \\ = \frac{sin {}^{2} x}{1 - \frac{cos \: x}{sin \: x} } + \frac{cos {}^{2} x}{1 - \frac{sin \: x}{cos \: x} } \\ \\ = \frac{ \frac{sin {}^{2} x}{sin \: x - cos \: x} }{sin \: x} + \frac{ \frac{cos {}^{2}x }{cos \: x - sin \: x} }{cos \: x} \\ \\ = \frac{sin {}^{3}x }{sin \: x - cos \: x} + \frac{cos {}^{3}x }{ - (sin \: x - cos \: x)} \\ \\ = \frac{sin {}^{3} x}{sin \: x - cos \: x} - \frac{cos {}^{3} x}{sin \: x - cos \: x} \\ \\ = \frac{sin {}^{3} x - cos {}^{3}x }{sin \: x - cos \: x} \\ \\ = \frac{(sin \: x - cos \: x)(sin {}^{2} x + cos {}^{2}x + sin \: x.cos \: x) }{sin \: x - cos \: x} \\ \\ = sin {}^{2} x + cos {}^{2} x + sin \: x.cos \: x \\ \\ = 1 + sin \: x.cos \: x \\ \\ = RHS \\ \\ hence \: proved$