Question

Prove that cat : R - {kπ | k ∈ Z} —> R is decreasing in all quadrants.

Answer 1

we have to prove that cot : R - {kπ | k ∈ Z}----> R is decreasing in all quadrants.

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) = cotx

differentiating f(x) with respect to x,

f'(x) = - cosec²x

as we know, square of any term always will be positive.

so, cosec²x > 0 for all x ∈ R-{kπ | k ∈ Z}

then, -cosec²x < 0 for all x ∈ R-{kπ | k ∈ Z}

hence, f'(x) < 0 for all x ∈ R-{kπ | k ∈ Z}

from theorem, f(x) is decreasing on R -{kπ | k ∈ Z}

hence, f(x) is decreasing in all quadrants {i.e., (0, π/2], (π/2, π), (π, 3π/2], (3π/2, 2π)]