Prove the identity:

\[ \cos(3x) + \cos(x) = 2\cos(2x)\cos(x) \]

cos α + cos β = 2 ⋅ cos 2 α + β ​ ⋅ cos 2 α − β ​ − − − − − − − − − − − − − − − − − α = 3 x an d β = x
L = cos α + cos β = cos ( 3 x ) + cos ( x ) ⇒ 2 ⋅ cos 2 α + β ​ ⋅ cos 2 α − β ​ = 2 ⋅ cos 2 3 x + x ​ ⋅ cos 2 3 x − x ​ = . = 2 ⋅ cos 2 4 x ​ ⋅ cos 2 2 x ​ = . = 2 ⋅ cos ( 2 x ) ⋅ cos ( x ) = R

Answered by kate200468 | 2024-06-24

To prove the identity cos ( 3 x ) + cos ( x ) = 2 cos ( 2 x ) cos ( x ) , we will use the sum-to-product identities and the double-angle formula for cosine.
First, recall the sum-to-product identities:
cos A + cos B = 2 cos ( 2 A + B ​ ) cos ( 2 A − B ​ )
Now, let's apply this identity to the left-hand side of our given identity:
cos ( 3 x ) + cos ( x ) = 2 cos ( 2 3 x + x ​ ) cos ( 2 3 x − x ​ )
Simplify the terms inside the cosine functions:
= 2 cos ( 2 4 x ​ ) cos ( 2 2 x ​ ) = 2 cos ( 2 x ) cos ( x )
This is exactly the right-hand side of the identity we wanted to prove. Therefore, we have shown that:
cos ( 3 x ) + cos ( x ) = 2 cos ( 2 x ) cos ( x )
Hence, the identity is proven.

Answered by ArpitaK | 2024-06-26

We proved the identity cos ( 3 x ) + cos ( x ) = 2 cos ( 2 x ) cos ( x ) using a trigonometric identity that relates the sum of cosines to products of cosines. By substituting and simplifying, we showed that both sides of the equation are equal. Hence, the identity holds true.
;

Answered by kate200468 | 2024-10-12