How would I prove tan(-x)=-tan(x)?

Question

How would I prove tan(-x)=-tan(x)?

Answer 1

sin(-z)/cos(-z)=-sin(z)/cos(z)=-tan(z)

Answer 2

tan^(-1)(tan(-x)) = -tan^(-1)(tan(x))

-x = -x

Answer 3

By convention, the angle -x (assuming an acute angle) places it into the fourth quadrant.  In that quadrant only cosine is positive.  Therefore, tan (-x) is negative.  Tan (x) is positive and -tan(x) is negative.

Since both -tan(x) and tan(-x) are both negative, and the size of the angle is the same in each case, tan(-x) = -tan(x)

If x is obtuse, -x will be in the 3rd quadrant (where tan is positive) but x will be in the second quadrant (where tan is negative).  Therefore, tan (-x) is positive and -tan(x) is also positive.

The above thinking works for any angle x since between the sign of the angle, the sign of the expression and the sign of the ratio in the given quadrant, the statement tan(-x) = -tan(x) will always be equal FOR EVERY VALUE OF x.

NB  The above would be a lot easier to prove using diagrams but I hope it is clear enough for you.