1 - cos squared A. sec square b+ tan square b. 1 - sin square A
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Step-by-step explanation:
Given,
L.H.S. = (1 - cos^2 A ).sec^2 B + tan ^2 B (1 - sin^2 A )
= (sin ^2 A ) sec^2 B + tan ^2 B (cos^2 A )
= (sin^2 A ) (1 + tan ^2 B ) + tan ^2 B (cos ^2 A )
= sin ^2 A + sin^2 A. tan^2 B + tan ^2 B (cos ^2 A )
= sin^2 A + tan^2 B [ sin^2 A + cos ^2 A ]
= sin^2 A + tan^2 B (1) [∵sin^2 A + cos ^2 A = 1]
=sin^2 A + tan^2 B = R.H.S.
∴L.H.S. = R.H.S.
Hence (1 - cos^2 A ) .sec^2 B = sin^2 A + tan^2 B is proved.
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