If tan^-1(2x)+tan^-1(3x)=π /4, then x is:

A. 1/6. Step 1: tan^-12x+tan^-13x=tan^-15x1-6x^2=π /4, so 5x1-6x^2=1. Step 2: 6x^2+5x-1=0, i.e. (6x-1)(x+1)=0, so x=1/6 or x=-1. Step 3: x=-1 makes both angles negative (their sum is -3π /4), so x=1/6. Check: x=1/6 gives tan^-11/3+tan^-11/2, and 1/3+1/2/1-1/6=1, so the sum is π /4. Trap: (B) 1/3 (the source key) gives tan^-12/3+tan^-11, which is more than π /4 because tan^-11 alone is π /4.

Question 14 of 21 · Inverse Trigonometric Functions

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NIMCET 2009

If $t an^{- 1} (2 x) + t an^{- 1} (3 x) = \frac{π}{4}$ , then $x$ is:

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $\frac{1}{6}$

Detailed Solution

$t an^{- 1} 2 x + t an^{- 1} 3 x = t an^{- 1} \frac{5 x}{1 - 6 x ^{2}} = \frac{π}{4}$ , so $\frac{5 x}{1 - 6 x ^{2}} = 1$ .

$6 x^{2} + 5 x - 1 = 0$ , i.e. $(6 x - 1) (x + 1) = 0$ , so $x = \frac{1}{6}$ or $x = - 1$ .

$x = - 1$ makes both angles negative (their sum is $- \frac{3 π}{4}$ ), so $x = \frac{1}{6}$ .

✓ Check

x = \frac{1}{6}

gives

t an^{- 1} \frac{1}{3} + t an^{- 1} \frac{1}{2}

, and

\frac{1/3 + 1/2}{1 - 1/6} = 1

, so the sum is

\frac{π}{4}

⚠ Common trap

(B)

\frac{1}{3}

(the source key) gives

t an^{- 1} \frac{2}{3} + t an^{- 1} 1

, which is more than

\frac{π}{4}

because

t an^{- 1} 1

alone is

\frac{π}{4}

Quick tip

$\frac{5 x}{1 - 6 x ^{2}} = 1 \Rightarrow (6 x - 1) (x + 1) = 0 \Rightarrow x = \frac{1}{6}$

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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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