Question

Prove the following trigonometric identities. $\frac{tan^{2}A}{1+tan^{2}A}+\frac{cot^{2}A}{1+cot^{2}A}=1$

Answer 1

Answer with Step-by-step explanation:

Given :

tan²A/(1+ tan²A) + cot²A/(1 + cot²A) = 1

LHS : tan²A/(1 + tan²A) + cot²A/(1 + cot²A)

= (sin²A/cos²A) /[1 + (sin²A/cos²A) + + (cos²A/sin²A) / [(1 + (cos²A/sin²A)]  

[By using the identity, cotθ = cosθ/sinθ  ,  tanθ = sinθ/cosθ ]

= (sin²A/cos²A) /[(cos²A+ sin²A)/cos²A) + (cos²A/sin²A) / [(sin²A + cos²A)/sin²A)]  

= (sin²A/cos²A) × cos²A/(cos²A+ sin²A) + (cos²A/sin²A) × sin²A  / (sin²A + cos²A)

= [(sin²A /(cos²A+ sin²A)] + [(cos²A / (sin²A + cos²A)]

= (sin²A + cos²A)/(cos²A + sin²A)

= 1

L.H.S = R.H.S  

Hence Proved..

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Answer 2

Answer:

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