Relation f⁻¹(x) with f⁻¹(1/x) – 1
Theorem: sin⁻¹(1/x) = cosec⁻¹(x), for all \(x\in \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)\),
Proof: sin⁻¹(1/x) = cosec⁻¹(x)
Let cosec⁻¹(x) = θ
\(\theta \in \left[ -\frac{\pi }{2},\frac{\pi }{2} \right]-\left\{ 0 \right\}\),
And \(x\in \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)\),
x = cosecθ
1/x = 1/ cosecθ
1/x = sinθ
θ = sin⁻¹(1/x)
cosec⁻¹(x) = θ
cosec⁻¹(x) = sin⁻¹(1/x)
sin⁻¹(1/x) = cosec⁻¹(x), for all \(x\in\left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)\),
Hence proved
Examples:
1.sin⁻¹(1/2) = cosec⁻¹(2)
2.sin⁻¹(1/√2) = cosec⁻¹(√2)
3.sin⁻¹(√3/2) = cosec⁻¹(2/√3)