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

Sine double angle rule is used to expand double angle functions, such as $\sin{2x}$, $\sin{2A}$, $\sin{2\alpha}$, $\sin{2\theta}$ and etc. in terms of sin and cos of angles but the same sin double angle functions can also be expanded in terms of tangent of angle.

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

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

$(3) \,\,\,\,\,\,$ $\sin{2A}$ $\,=\,$ $\dfrac{2\tan{A}}{1+\tan^2{A}}$

$(4) \,\,\,\,\,\,$ $\sin{2\alpha}$ $\,=\,$ $\dfrac{2\tan{\alpha}}{1+\tan^2{\alpha}}$

Proof

The expansion of sin of double angle formula in terms of tan of angle can be derived on the basis of the expansion of sin double angle rule and some trigonometric identities.

Basic knowledge of sin double angle formula

Take theta as an angle and sin double angle is written as $\sin{2\theta}$. According to sin double angle identity, it can be expanded in terms of sin and cos of angles.

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

Now, it is time to eliminate sin and cos of angle terms to transform them into tan of angle.

Transformation of Sin in terms of Tan

Firstly, try to transform sin of angle in terms of tan of angle. It is possible, if sin of angle is divided by cos of angle. So, do a mathematically acceptable adjustment to transform sin of angle as tan of angle.

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

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

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

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

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

According to quotient identity of sine, cosine and tangent functions, the quotient of sin of angle by cos of angle is equal to tan of angle.

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

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

Transformation of Cos in terms of Tan

Now, convert the cos of angle in terms of tan of angle but it cannot be done directly. However, it can be done in two steps. Actually, cos function does not have direct relation with tan function but it has direct relation with secant function and secant function has direct relation with tan function.

So, convert cos of angle in terms of secant of angle by applying reciprocal identity of cosine and secant functions.

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

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

According to Pythagorean identity of secant and tangent functions, square of secant of angle can be converted as square of tan of angle.

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



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