[tex]\sin 2 \theta \sec \theta=2 \sin \theta[/tex]
Solution:
To prove that [tex]\sin 2 \theta \sec \theta=2 \sin \theta[/tex].
Let us take LHS which is equal to RHS.
[tex]LHS=\sin 2 \theta \sec \theta[/tex]
Using basic trigonometric identity: [tex]\sec (\theta)=\frac{1}{\cos (\theta)}[/tex]
[tex]$=\frac{1}{\cos \theta} \sin 2 \theta[/tex]
[tex]$=\frac{\sin 2 \theta}{\cos \theta}[/tex]
Using trigonometric identity: [tex]\sin (2 x)=2 \cos (x) \sin (x)[/tex]
[tex]$=\frac{2 \cos \theta \sin \theta}{\cos \theta}[/tex]
Cancel both cosθ in the numerator and denominator.
[tex]=2 \sin \theta[/tex]
[tex]=RHS[/tex]
LHS = RHS
[tex]\sin 2 \theta \sec \theta=2 \sin \theta[/tex]
Hence proved.
