Math Doubts

Proof of Cosine double angle identity in square of Sine

The cosine double angle identity can be expanded in terms of square of sine function as follows.

$\cos{2\theta}$ $\,=\,$ $1-2\sin^2{\theta}$

Now, it is your turn to learn how to prove the cos double angle identity in terms of sine squared function in trigonometry.

In mathematics, the sine in square form is written as $\sin^2{\theta}$ and cosine of double angle function is written as $\cos{2\theta}$ when the angle of a right triangle is denoted by theta.

Write the cos double angle identity

According to the cos double identity proof, the cosine of double angle is equal to the difference of the squares of sine and cosine of angle.

$\cos{2\theta}$ $\,=\,$ $\cos^2{\theta}-\sin^2{\theta}$

Convert the Cosine function into Sine

According to the cosine squared identity, the square of cos function can be written in terms of square of sin function.

$\implies$ $\cos{2\theta}$ $\,=\,$ $\Big(1-\sin^2{\theta}\Big)-\sin^2{\theta}$

Simplify the Trigonometric expression

Now, simplify the right hand side trigonometric expression of the equation for deriving the expansion of cos double angle formula in square of sine function.

$\implies$ $\cos{2\theta}$ $\,=\,$ $1-\sin^2{\theta}-\sin^2{\theta}$

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

Other forms

The angle in this cosine double angle identity can be represented by any symbol. However, it is popularly written in the following two forms. You can derive this trigonometric identity in terms of any symbol by the above steps.

$(1) \,\,\,\,\,\,$ $\cos{2x}$ $\,=\,$ $1-2\sin^2{x}$

$(2) \,\,\,\,\,\,$ $\cos{2A}$ $\,=\,$ $1-2\sin^2{A}$

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