A mathematical relation of tangent of negative angle with tangent of positive angle is called tan of negative angle identity.
$\tan{(-\theta)} \,=\, -\tan{\theta}$
The mathematical relation between tangent of negative angle and tan of positive angle is derived in mathematical form by geometrical method in trigonometry.
$\Delta POQ$ is a right angled triangle and take its positive angle is denoted by theta. Express tan of positive angle in terms of ratio of lengths of the two respective sides.
$\tan{\theta} \,=\, \dfrac{PQ}{OQ}$
The $\Delta POQ$ is constructed in first quadrant. Therefore, the lengths of both adjacent and opposite sides are positive and take they both are represented by $x$ and $y$ respectively.
$\implies$ $\tan{\theta} \,=\, \dfrac{y}{x}$
Construct the same triangle with negative angle. Therefore, the angle of $\Delta QOR$ is negative theta, denoted by $–\theta$.
On the basis of this data, write the tan of negative angle in terms of ratio of lengths of respective sides.
$\tan{(-\theta)} \,=\, \dfrac{QR}{OQ}$
Due to construction of the triangle with negative angle, geometrically the length of opposite side will be $–y$ but the length of adjacent side is same.
$\implies$ $\tan{(-\theta)} \,=\, \dfrac{-y}{x}$
Now, compare equations of both tangent of positive angle and tan of negative angle.
$\tan{\theta} \,=\, \dfrac{y}{x}$
$\tan{(-\theta)} \,=\, -\dfrac{y}{x}$
The two equations disclose that tan of negative angle is always equal to negative of tangent of angle.
$\,\,\, \therefore \,\,\,\,\,\,$ $\tan{(-\theta)} \,=\, -\tan{\theta}$
It is called tangent of negative angle identity and used as a formula in trigonometric mathematics.
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