Question

Prove that 1-cos2A/sin2A =TanA

Answer 1

Step-by-step explanation:

To prove that:

$\frac{1-\cos 2A}{\sin2 A}=\tan A$

Let us take left hand side and prove it.

$LHS = \frac{1-\cos 2 A}{\sin 2 A}$

Using the trigonometric identity: $\cos (2 x)=1-2 \sin ^{2}(x)$

         $= \frac{1-(1-2\sin^2A)}{\sin 2 A}$

Using the trigonometric identity: $\sin (2 x)=2 \cos (x) \sin (x)$

        $= \frac{1-1+2\sin^2A}{2 \sin A \cos A }$

        $= \frac{2\sin^2A}{2 \sin A \cos A }$

        $= \frac{2\sin A \sin A }{2 \sin A \cos A }$

Cancel the common term 2 sin A on both numerator and denominator.

         $=\frac{\sin A}{\cos A}$  

Using the trigonometric identity: $\frac{\sin (x)}{\cos (x)}=\tan (x)$

         = tan A

         = RHS

$\frac{1-\cos 2A}{\sin2 A}=\tan A$

Hence proved.