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$\sin 2x = 2 \sin x \cos x$

$= \dfrac{2 \sin x \cos x}{\cos x \sec^2 x} \cdot \cos x \sec^2 x$

$= 2 \cdot \dfrac{\sin x}{\cos x} \cdot \dfrac{1}{\sec^2 x} \cdot \cos x \cdot \cos x \sec^2 x$

$= 2 \cdot \tan x \cdot \dfrac{1}{\sec^2 x} \cdot \cos^2 x \sec^2 x$

$= 2 \cdot \tan x \cdot \dfrac{1}{1 + \tan^2 x} \cdot \cos^2 x \cdot \dfrac{1}{\cos^2 x}$

In 2sinxcosx/cosxsec^2x•cosxsec^2x, why is sec^2x is used?

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