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  4. If tan2 A = 2 tan2 B + 1, then cos 2A + sin2B equals

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Q. If $\tan^2 \, A = 2 \, \tan^2 \, B + 1$, then $\cos \, 2A + \sin^2B$ equals

Trigonometric Functions

Solution:

Given : $\tan^{2} A = 2 \tan^{2 } B + 1$
$ \Rightarrow 1+ \tan^{2}A = 2 \tan^{2} B + 1 + 1$
$ \Rightarrow \sec^{2}A = 2 \sec^{2} B$
$ \Rightarrow \cos^{2 }B = 2 \cos^{2} A$
$ \Rightarrow \cos^{2}B = 1 + \cos2 A $
$ \Rightarrow \cos^{2} B - 1= \cos2 A $
$ \Rightarrow - \sin^{2} B = \cos2 A $
$ \Rightarrow \cos2 A + \sin^{2} B = 0 $