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

c)

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)=(-\sin(y))/(\cos(y))=(\sin(-y))/(\cos(y))=(\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`