# Reciprocal Arguments and Inverse Sum Identities

## Reciprocal Arguments and Inverse Sum Identities

Reciprocal Arguments:

(i) sin⁻¹ (1/x) = cosec⁻¹ x, ∀ x ϵ (-∞, -1] ∪ [1, ∞),

(ii) cos⁻¹ (1/x) = sec⁻¹ x, ∀ x ϵ (-∞, -1] ∪ [1, ∞),

(iii) {{\tan }^{-1}}\left( \frac{1}{x} \right)=\left\{\begin{align} & {{\cot }^{-1}}x,\ \forall \ \ x>0 \\ & -\pi +{{\cot }^{-1}}x,\ \forall \ x<0 \\ \end{align} \right\}.

Inverse Sum Identities:

(i) sin⁻¹ x + cos⁻¹ x = π/ 2, ∀ x ϵ [-1, 1],

(ii) tan⁻¹ x + cot⁻¹ x = π/ 2, ∀ x ϵ R,

(iii) sec⁻¹ x + cosec⁻¹ x = π/ 2, ∀ x ϵ (-∞, -1] ∪ [1, ∞).

Example: sin⁻¹ (1 – x) – 2 sin⁻¹ x = π/2, then x is equal to

Solution: Given that,

sin⁻¹ (1 – x) – 2 sin⁻¹ x = π/2

– 2 sin⁻¹ x = – sin⁻¹ (1 – x) + π/2

– 2 sin⁻¹ x = cos⁻¹ (1 – x)

(∵ sin⁻¹ (1 – x) + cos⁻¹ (1 – x) = π/2)

Multiply both sides by cos

cos (- 2 sin⁻¹ x) = cos [cos⁻¹ (1 – x)]

cos (- 2 sin⁻¹ x) = (1 – x)

cos (2 sin⁻¹ x) = (1 – x)

(∵ cos(-x) = cos x)

1 – 2 sin² (sin⁻¹ x) = (1 – x)

1 – 2 [sin (sin⁻¹ x)]² = (1 – x)

1 – 2x² = 1 – x

2x² – x =0

x (2x – 1) = 0

x = 0 (or) ½

But x = ½ does not satisfy the given equation, So x = 0.