Question

Prove that cosec is an increasing function in(π/2,π).

Answer 1

we have to prove that cosec is an increasing function in (π/2, π).

Theorem : Let f be a differentiable real function defined on an open interval (a,b).

• (i) If f’(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b)
• (ii) If f’(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b).

here, f(x) = cosecx

differentiating both sides,

f'(x) = -cosecx.cotx

in interval x ∈ (π/2, π) [ i.e., 2nd quadrant] ;

cosecx will be positive and cotx will be negative.

so, cosecx. cotx will be negative.

then, -cosecx.cotx will be positive. i.e., f'(x) = -cosecx.cotx > 0 for x ∈ (π/2, π).

hence, f(x) = cosecx is increasing function in (π/2, π).