Question

plz help me I'm struck here​

Answer 1

$LHS =\red{ \frac{cot^{3}\theta sin^{3}\theta}{(cos\theta+sin \theta)^{2} }+ \frac{tan^{3}\theta cos^{3}\theta}{(cos\theta+sin \theta)^{2}}}$

$= \frac{\frac{cos^{3}\theta }{sin^{3} \theta } \times sin^{3}\theta}{(cos\theta+sin \theta)^{2} }+ \frac{\frac{sin^{3}\theta }{cos^{3} \theta} \times cos^{3}\theta}{(cos\theta+sin \theta)^{2}}$

$= \frac{cos^{3}\theta }{(cos\theta+sin \theta)^{2} }+ \frac{sin^{3}\theta }{(cos\theta+sin \theta)^{2}}$

$= \frac{cos^{3}\theta + sin^{3}\theta }{(cos\theta+sin \theta)^{2}}$

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

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

$On\: Dividing\: numerator\: and \\denominator\: by \: sin \theta cos \theta, we \:get$

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

$= \green {\frac{ sec \theta Cosec \theta - 1}{ Cosec \theta + sec \theta } }\\= RHS$

$Hence\: proved$

•••♪

Answer 2

$\huge{\purple{\bigstar{ANSWER}}}$