If $s in^{- 1} x + cos^{- 1} (1 - x) = s in^{- 1} (1 - x)$ , then $x$ satisfies the equation:

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — None of these

Detailed Solution

Domain: both $x$ and $1 - x$ must lie in $[- 1, 1]$ , so $0 \leq x \leq 1$ .

Let $α = s in^{- 1} (1 - x)$ , so $cos^{- 1} (1 - x) = \frac{π}{2} - α$ . The equation becomes $s in^{- 1} x = 2 α - \frac{π}{2}$ .

Taking sine: $x = - cos 2 α = 2 s in^{2} α - 1 = 2 (1 - x)^{2} - 1$ , i.e. $2 x^{2} - 5 x + 1 = 0$ . Its root in $[0, 1]$ is $x = \frac{5 - 17}{4} \approx 0.219$ , and it satisfies none of the equations in (A), (B), (C).

✓ Check

For

x = 0.219

s in^{- 1} 0.219 + cos^{- 1} 0.781 \approx 0.221 + 0.675 = 0.896

, and

s in^{- 1} 0.781 \approx 0.896

⚠ Common trap

(B)

2 x^{2} - 3 x = 0

(the source key) gives

x = 0

\frac{3}{2}

. At

x = 0

the left side is

0 + cos^{- 1} 1 = 0

but the right side is

s in^{- 1} 1 = \frac{π}{2}

, and

\frac{3}{2}

is outside the domain.

Quick tip

$α = sin^{- 1} (1 - x) : x = - cos 2 α = 2 (1 - x)^{2} - 1 \Rightarrow 2 x^{2} - 5 x + 1 = 0$

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Inverse Trigonometric Functions — Preparation

This question tests your understanding of Inverse Trigonometric Functions, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Inverse trigonometric functions, such as sin^-1 x, cos^-1 x and tan^-1 x, return the angle whose trigonometric ratio is a given value. Because trigonometric functions repeat, each inverse is defined only on a principal value branch. For example, sin^-1 x takes values between -pi/2 and pi/2. This topic covers domains and principal value ranges, finding principal values, and key properties such as sin^-1 x + cos^-1 x = pi/2 and the formulas for tan^-1 x + tan^-1 y. Exam questions test simplifying expressions, solving equations, and evaluating compositions such as sin(cos^-1 x). The most common mistake is ignoring the principal range, for example assuming sin^-1(sin x) always equals x. Practise the inverse trigonometric functions questions below, each with a step-by-step solution.

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