Math Doubts

Cosine of Double angle in terms of Cos of angle

Formula

$\large \cos 2\theta = 2\cos^2 \theta -1$

Proof

According to the double angle trigonometric identity, the cosine of double angle is equal to the subtraction of square of sine of angle from square of cosine of angle.

$\cos 2\theta = \cos^2 \theta -\sin^2 \theta$

The square of sine of angle can be transformed in terms of square of cosine of angle by the Pythagorean trigonometric identity of sine and cosine functions.

$\implies \cos 2\theta$ $=$ $\cos^2 \theta -(1-\cos^2 \theta)$

$\implies \cos 2\theta$ $=$ $\cos^2 \theta -1+\cos^2 \theta$

$\implies \cos 2\theta$ $=$ $\cos^2 \theta +\cos^2 \theta -1$

$\therefore \,\,\,\,\,\,$ $\cos 2\theta = 2\cos^2 \theta -1$

Therefore, the cosine of double angle in terms of cosine of angle can be expanded as the subtraction of number one from the twice the square of cosine of angle.

Other form

The trigonometric property is written as follows if the angle of the right angled triangle is $x$.

$\cos 2x = 2\cos^2 x -1$

It states that the angle of right angled triangle can be denoted by any symbol, the expansion of cosine of double angle in terms of cosine of angle is in this form.



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