Given: \(\theta=\frac{a}{b}\)
To Prove: a sin 2θ + b cos 2θ = b
Given: \(\theta=\frac{a}{b}\)
We know that,
By Pythagoras Theorem,
(Perpendicular)2 + (Base)2 = (Hypotenuse)2
⇒ (a)2 + (b)2 = (H)2
⇒ a2 + b2 = (H)2
⇒ H = \(\sqrt{a^2+b^2}\)
So,
Taking LHS,
= a sin 2θ + b cos 2θ
We know that,
sin 2θ = 2 sin θ cos θ
and cos 2θ = 1 – 2 sin2θ
= a(2 sin θ cos θ) + b(1 – 2 sin2θ)
Putting the values of sinθ and cosθ, we get
= b
= RHS
∴ LHS = RHS
Hence Proved
