Question

If –1 ≤ x ≤ 1, the prove that sin^–1 x + cos^–1 x = `π/2`

Answer 1

Let sin^–1 x = θ, where x ∈ [–1, 1] and `θ ∈ [-π/2, π/2]`

∴ `- θ ∈ [-π/2, π/2]`

∴ `π/2 - θ ∈ [0, π]`, the principal domain of the cosine function.

∴ `cos(π/2 - θ)` = sin θ

`cos(π/2 - θ)` = x

∴ cos^–1 x = `π/2 - θ`

∴ `θ + cos^-1x = π/2`

∴ sin^–1 x + cos^–1 x = `π/2`