Question

Prove that 1-cos2A/sin2A =TanA

Answer 1

Explanation:

To prove that:

`$(1-\cos 2A)/(\sin2 A)=\tan A`

Let us take left hand side and prove it.

`$LHS = (1-\cos 2 A)/(\sin 2 A)`

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

`$= (1-(1-2\sin^2A))/(\sin 2 A)`

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

`$= (1-1+2\sin^2A)/(2 \sin A \cos A )`

`$= (2\sin^2A)/(2 \sin A \cos A )`

`$= (2\sin A \sin A )/(2 \sin A \cos A )`

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

`$=(\sin A)/(\cos A)`

Using the trigonometric identity:
`(\sin (x))/(\cos (x))=\tan (x)`

= tan A

= RHS

`$(1-\cos 2A)/(\sin2 A)=\tan A`

Hence proved.