Question
Question: If \[{{\cos }^{2}}\theta -{{\sin }^{2}}\theta ={{\tan }^{2}}\phi \]. Prove that \[\cos \phi =\dfrac{...
If cos2θ−sin2θ=tan2ϕ. Prove that cosϕ=2cosθ1.
Solution
In order to prove cosϕ=2cosθ1, firstly we will be applying the trigonometric identities. And then for easy calculation and to bring it to the form of identities, we will be adding one to both sides of the equation. And solving them accordingly will give us the required answer.
Complete step-by-step answer:
Now let us have a brief regarding the trigonometric identities. Generally, trigonometric identities are the equalities that occur for the trigonometric ratios and are true for all of the occurring variables. These trigonometric identities contain functions of one or more than one angles. There are six trigonometric ratios, they are sine, cosine, tangent, cotangent, secant and cosecant. These ratios are generally expressed as the ratios of the sides of a right angled triangle.
Now let us prove that if cos2θ−sin2θ=tan2ϕ, then cosϕ=2cosθ1.
We are given with cos2θ−sin2θ=tan2ϕ.
So consider cos2θ−sin2θ=tan2ϕ
Now let us add 1 on both sides of the equation. Then we get,
cos2θ−sin2θ+1=tan2ϕ+1
Upon rearranging the terms in order to obtain the identity, we get
cos2θ+(1−sin2θ)=(tan2ϕ+1)
We know that \left( 1-{{\sin }^{2}}\theta \right)$$$$=\cos \theta and also \left( {{\tan }^{2}}\phi +1 \right)$$$$={{\sec }^{2}}\phi
On replacing these values we get,