Cosine of Double angle in terms of square of Sine of angle

Formula

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

Proof

As per the double angle identity of cosine function in trigonometry, the cosine of double angle can be expanded as the subtraction of square of sine of angle from the square of the cosine of angle.

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

Use the Pythagorean identity of sine and cosine to convert square of cosine of angle in terms of sine squared angle.

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

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

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

The trigonometric expression expresses the expansion of the cosine of double angle in terms of sine of angle. In other words, the cos of double angle is equal to subtraction of twice the square of sine of angle from one.

Other form

If the angle of the right angled triangle is $x$, it can be expressed in same form but in terms of $x$.

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

The angle of the right angled triangle can be represented by any symbol in trigonometry, the expansion of cosine of double angle in terms of sine of angle is in this mathematical form.

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