Question

What is the period of the function f(x) = ln(2 + sin²x)?
1. \(\rm \frac{\pi}{2}\)
2. π
3. 2π
4. 3π

Answer 1

Correct Answer - Option 2 : π

Formula used:

cos2θ = 1 - 2sin²θ 

Calculation:

f(x) = ln(2 + sin2x)

⇒ f(x) = ln[2 + (1 - cos2x)/2]

⇒ f(x) = ln[(5 - cos2x)]/2      ----(i)

If we put (x + π) in place of x, we get

⇒ f(x + π) = ln[(5 - cos2(x + π)/2]

⇒ f(x + π) = ln[5 - cos(2π + 2x)]/2 

⇒ f(x + π) = ln[(5 - cos2x)]/2      ----(ii) 

From (i) and (ii), we get 

f(x + π) = f(x)

∴ The period of the function f(x) = ln(2 + sin2x) is π.