Question

Sin2xsecx = 2sinx

How do you prove this is an identity? ​

Answer 1

`\sin 2 \theta \sec \theta=2 \sin \theta`

Solution:

To prove that
`\sin 2 \theta \sec \theta=2 \sin \theta`.

Let us take LHS which is equal to RHS.

`LHS=\sin 2 \theta \sec \theta`

Using basic trigonometric identity:
`\sec (\theta)=(1)/(\cos (\theta))`

`$=(1)/(\cos \theta) \sin 2 \theta`

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

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

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

Cancel both cosθ in the numerator and denominator.

`=2 \sin \theta`

`=RHS`

LHS = RHS

`\sin 2 \theta \sec \theta=2 \sin \theta`

Hence proved.