Sin2xsecx = 2sinx How do you prove this is an identity?

Question

asked Nov 20, 2021 188k views

1 Answer

Ryne Everett · Answer 1 · 2021-11-26T13:03:08+0000

$\sin 2 \theta \sec \theta=2 \sin \theta$

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.