Relation f⁻¹(x) with f⁻¹(1/x) – 1

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)