Proof of Sin double angle identity in Tan function

The sine double angle identity can be expanded in terms of tangent as the following rational form.

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

Now, let learn how to derive the sine double angle rule in terms of tan function in trigonometry.

Let $x$ represents an angle of a right triangle, the sine, cosine and tan of angle are written as $\sin{x}$, $\cos{x}$ and $\tan{x}$ respectively. Similarly, the sine of double angle is written as $\sin{2x}$ in trigonometric mathematics.

Write the sin double angle identity

We have proved the sin double angle identity geometrically in mathematical form as follows.

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

Convert the Sine function into Tangent

Try to transform the sine in the right hand side of the equation in tan function acceptably. Actually, it can be done by multiplying and diving the expression with cosine.

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

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

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

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

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

On the basis of the sine by cosine quotient identity, the quotient of $\sin{x}$ by $\cos{x}$ can be written as $\tan{x}$.

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

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

Convert the Cosine function into Tangent

According to the basic trigonometric identities, it is not possible express the cosine in terms of tan functions. However, it can be transformed in terms of secant by the reciprocal identity of secant.

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

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

Now, express the 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 the tan of angle by the sum of one and square of tan of angle.

Other forms

The angle in the sine double angle identity can be represented by any symbol. Actually, the sine double angle formula in tan function is also popularly written in two other forms as follows. So, you can derive it in any form by the above procedure.

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

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