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If tan^2B=sin^2A-cos^2A then prove that tan^2A=sin^2B-cos^B

If tan^2B=sin^2A-cos^2A then prove that tan^2A=sin^2B-cos^B

Grade:11

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student

you did mistake in typing question ,it should be like this
cos²A - sin²A = tan²B , then prove , cos²B - sin²B = tan²A

Given, cos²A - sin²A = tan²B
⇒cos²A - ( 1 - cos²A ) = tan²B [∵ sin²Ф = 1 - cos²Ф ]
⇒ cos²A - 1 + cos²A = tan²B
⇒ 2cos²A - 1 = tan²B
⇒ 2cos²A = 1 + tan²B
⇒ 2cos²A = sec²B [ ∵ sec²Ф = 1 + tan²Ф ]
⇒2cos²A.cos²B =1 [ ∵ secФ = 1/cosФ ]
⇒2cos²B = sec²A
= cos²B + cos²B = 1 + tan²A
⇒ cos²B + cos²B - 1 = tan²A
⇒cos²B - (1 - cos²B) = tan²A
⇒ cos²B - sin²B = tan²A
Hence proved

Thanks

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