Cos Double Angle Formula
Trigonometry is a branch of mathematics that deals with the study of the relationship between the angles and sides of a rightangled triangle. Cosine is one of the primary trigonometric ratios which helps in calculating the ratio of base and hypotenuse. So, cos can be defined as the ratio of the length of the base to that of the length of the hypotenuse in a rightangled triangle.
Let us learn the cos double angle formula with its derivation and a few solved examples.
What is Cos Double Angle Formula?
We will use the formula of cos(A + B) to derive the cos double angle formula.
Formula 1:
The double angle identity of cos x can be obtained by using the sum formula of cos (A + B).
We have cos (A + B) = cos A cos B  sin A sin B  (1)
Substitute x for both A and B in Equation (1).
cos (A + B) = cos A cos B  sin A sin B
cos (x + x) = cos x cos x sin x sin x
cos (2x) = cos^{2} x  sin^{2} x
cos (2x) = cos^{2} x  sin^{2} x
Formula 2:
Now, we use the trigonometric identity cos^{2} x + sin^{2} x = 1 to derive another cos double angle formula.
Substitute 1  cos^{2} x for sin^{2} x in Formula 1.
cos (2x) = cos^{2} x  sin^{2} x
= cos^{2} x  (1  cos^{2} x)
= 2cos^{2 }x  1
cos (2x) = 2cos^{2 }x  1
Formula 3:
Again we will use the trigonometric identity cos^{2} x + sin^{2} x = 1 to derive one more cos double angle formula.
Substitute 1  sin^{2} x for cos^{2} x in formula 1.
cos (2x) = cos^{2} x  sin^{2} x
= 1  sin^{2} x  sin^{2} x
= 1  2sin^{2} x
cos (2x) = 1  2sin^{2} x
Let's take a quick look at a couple of examples to understand the cos double angle formula better.
Solved Examples on Cos Double Angle Formula

Example 1: If sin x = 4/5, then find the value of cos (2x) using the cos double angle formula.
Solution:
Substitute 4/5 for sin x in the formula cos (2x) = 1  2sin^{2} x.
cos (2x) = 1  2sin^{2} x
= 1  2 (4/5)^{2}
= 1  32/25
= 7/25 
Example 2: If cos x = ⎷5/3, then find the value of cos (2x) using the cos double angle formula.
Solution:
Substitute ⎷5/3 for cos x in the formula cos (2x) = 2cos^{2 }x  1.
cos (2x) = 2cos^{2 }x  1
= 2 (⎷5/3)^{2}1
= 10/9  1
= 1/9
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