Question

Assertion :Let f be any one of the six trigonometric functions.Let $A,BϵR$satisfying $f\left(2A\right)=f\left(2B\right).$ Statement 1: $A=n\pi +B$ for some $nϵZ.$ Reason: Statement 2: $2\pi$ is one of the period of f

• A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is A Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1

$f\left(2A\right)$

$=f(2n\pi +2B)$

$=f\left(2B\right)$

Now by definition of a periodic function

$f\left(a\right)=f(T+a)$ where T is the period of the function.

Here $f(2n\pi +2B)=f\left(2B\right)$

Or

$f(2n\pi +\alpha )=f\left(\alpha \right)$

Hence $f\left(x\right)$ has a period of $2\pi$.