(sin2A)=(2tanA)/(sec^2A)

prove the identity

2tanA /(sec^2A)

=[ 2 sin A / cos A ] / [ 1/ cos ^2 A]

=[ 2 sin A / cos A ] cos ^2 A

= 2 sin A cos A
= sin 2A
proved

1 - sin 2x
= cos^2 x + sin^2 x - 2 sin x cos x
= (cos x - sin x) ^2
proves

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