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Prove that: { { } }{ ( 1- { } ) } + { { } }{ ( 1-

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Prove that: $\frac{sin θ}{( 1 - cot θ )} + \frac{cos θ}{( 1 - tan θ )} = cos θ + sin θ$

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Prove that:

\frac{sin θ}{( 1 - cot θ )} + \frac{cos θ}{( 1 - tan θ )} = cos θ + sin θ

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Text solutionVerified

L H S = \frac{sin θ}{1 - cot θ} + \frac{cos θ}{1 - tan θ}

= \frac{sin θ}{1 - ( 1/ tan θ )} + \frac{cos θ}{1 - tan θ}

= \frac{sin θ tan θ}{tan θ - 1} + \frac{cos θ}{1 - tan θ}

= \frac{sin θ tan θ - cos θ}{tan θ - 1}

= \frac{\frac{sin ^{2} θ}{cos θ} - cos θ}{tan θ - 1}

= \frac{sin ^{2} θ - cos ^{2} θ}{cos θ ( tan θ - 1 )}

= \frac{( sin θ + cos θ ) ( sin θ - cos θ )}{cos θ ( sin θ / cos θ - 1 )}

= \frac{( sin θ + cos θ ) ( sin θ - cos θ )}{sin θ - cos θ )}

= sin θ + cos θ

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Question Text	Prove that: $\frac{sin θ}{( 1 - cot θ )} + \frac{cos θ}{( 1 - tan θ )} = cos θ + sin θ$
Answer Type	Text solution:1
Upvotes	150