Question

Which of the following is an equivalent expression for tan to the power of minus 1 end exponent x plus tan to the power of minus 1 end exponent left parenthesis minus x right parenthesis
tan^-1x+tan^-1(-x)

a. -1

b. rad

c. 0

d. rad/2

e. 1

Answer 1

Answer:

c)

Step-by-step explanation:

To solve this, remember some properties of trigonometric functions.

Let $y=\tan^{-1} (x)$. Then, by definition of inverse function, $\tan(y)=x$. Multiply by -1 to both sides of this equation to get $-\tan(y)=-x$.

Note that  $-\tan(y)=\frac{-\sin(y)}{\cos(y)}=\frac{\sin(-y)}{\cos(y)}=\frac{\sin(-y)}{\cos(-y)}=\tan(-y)$ because sine is an odd function and cosine is an even function. Then $\tan(-y)=-x$. Take the inverse tangent in both sides to get  $-y=\tan^{-1} (-x)$.

Using the previous equations, we obtain:

$\tan^{-1} (x)+\tan^{-1} (-x)=y+(-y)=0$