Proof of Sin double angle formula in terms of Tan of angle

The sin double angle identity can also be expanded in terms of tan of angle. For example, if theta is angle of a right triangle, then sin double angle is written as $\sin{(2\theta)}$ and it is expanded in terms of tan of angle as the following mathematical form.

$\sin{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

In this way, the sin of double angle function can be expanded in terms of tan of corresponding angle and now, you have to learn how it is derived in trigonometry before going to use it as a formula in mathematics.

Proof

If x is used to represent angle of a right triangle, the sin of double angle is expressed as $\sin{(2x)}$ in mathematics.

Write expansion of sin double angle function

According to sin double angle identity, the $\sin{(2x)}$ function is expanded in terms of $\sin{x}$ and $\cos{x}$ functions.

$\sin{2x}$ $\,=\,$ $2\sin{x}\cos{x}$

Convert sin function into tan function

The $\sin{x}$ function can be converted as tanx function by dividing it by $cos{x}$ function.

$\implies \sin{2x}$ $\,=\,$ $2\sin{x}\cos{x} \times 1$

$\implies \sin{2x}$ $\,=\,$ $2\sin{x}\cos{x} \times \dfrac{\cos{x}}{\cos{x}}$

$\implies \sin{2x}$ $\,=\,$ $\dfrac{2\sin{x}\cos{x} \times \cos{x}}{\cos{x}}$

$\implies \sin{2x}$ $\,=\,$ $\dfrac{2\sin{x}\cos^2{x}}{\cos{x}}$

$\implies \sin{2x}$ $\,=\,$ $2 \times \dfrac{\sin{x}}{\cos{x}} \times \cos^2{x}$

According ratio identity of sin and cos functions, the quotient of them is $\tan{x}$.

$\implies \sin{2x}$ $\,=\,$ $2 \times \tan{x} \times \cos^2{x}$

$\implies \sin{2x}$ $\,=\,$ $2\tan{x} \times \cos^2{x}$

Transform cos function in terms of tangent

In fact, the cos function does not have any direct relation with tan function but it has direct relation with secant function as per reciprocal identity of secant function and the secant function has direct relation with tan function as per Pythagorean identity.

According to reciprocal identity of cos function, write square of cos function in its reciprocal form.

$\implies \sin{2x}$ $\,=\,$ $2\tan{x} \times \dfrac{1}{\sec^2{x}}$

$\implies \sin{2x}$ $\,=\,$ $\dfrac{2\tan{x}}{\sec^2{x}}$

Now, express square of secant function in terms of square of tan function as per secant squared identity.

$\,\,\, \therefore \,\,\,\,\,\, \sin{2x}$ $\,=\,$ $\dfrac{2\tan{x}}{1+\tan^2{x}}$

Therefore, it is proved that the expansion of sin of double angle is equal to quotient of two times tan of angle by the sum of one and square of tan of angle.

Example

The expansion of sin double angle identity in terms of tan of angle is used as a formula in trigonometry to expand any sin function which contains double angle.

$(1) \,\,\,\,\,\,$ $\sin{2A}$ $\,=\,$ $\dfrac{2\tan{A}}{1+\tan^2{A}}$

$(2) \,\,\,\,\,\,$ $\sin{2\alpha}$ $\,=\,$ $\dfrac{2\tan{\alpha}}{1+\tan^2{\alpha}}$

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