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Let f(x)=cos^(-1)((1-tan^(2)(x)/(2))/(1+...

Let `f(x)=cos^(-1)((1-tan^(2)(x)/(2))/(1+tan^(2)(x)/(2)))`. Then which of the following statement is/are true ?

A

Ranges of f(x) is `[0,pi)`

B

`f(x)=pi` has infinite roots

C

y = f(x) is identical with `y = cos^(-1)(cos x)`

D

y = f(x) has period `2pi`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`f(x)=cos^(-1)((1-"tan"^(2)(x)/(2))/(1+"tan"^(2)(x)/(2)))`
`=cos^(-1)(cos x), x in R - {x|x=(2n+1)pi, n in Z}`
Graph of the functions is as shown in the following figure.

From the graph, range of f(x) is `[0,pi)`
Also, f(x) is periodic with period `2pi`.
Clearly `f(x)=pi` has no real roots.
y = f(x) is not identical with `y=cos^(-1)(cos x)`
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