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.