If cos(α +β )=4/5 and sin(α -β )=5/13, where 0<α ,β <π /4, then tan2α is equal to

A. 56/33. Step 1: 0<α +β <π /2, so sin(α +β )=3/5 and tan(α +β )=3/4. Step 2: -π /4<α -β <π /4, so cos(α -β )=12/13 and tan(α -β )=5/12. Step 3: tan2α =3/4+5/121-3/4⋅5/12=14/12/33/48=56/33. Check: α +β 0.6435 and α -β 0.3948 give 2α 1.0383, and tan1.0383 1.697=56/33. Trap: (D) 33/56 is the reciprocal, i.e. cot2α, from inverting the final fraction.

Question 14 of 47 · Trigonometric Functions & Identities

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If $cos (α + β) = \frac{4}{5}$ and $s in (α - β) = \frac{5}{13}$ , where $0 < α, β < \frac{π}{4}$ , then $t an 2 α$ is equal to

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $\frac{56}{33}$

Detailed Solution

$0 < α + β < \frac{π}{2}$ , so $s in (α + β) = \frac{3}{5}$ and $t an (α + β) = \frac{3}{4}$ .

$- \frac{π}{4} < α - β < \frac{π}{4}$ , so $cos (α - β) = \frac{12}{13}$ and $t an (α - β) = \frac{5}{12}$ .

$t an 2 α = \frac{\frac{3}{4} + \frac{5}{12}}{1 - \frac{3}{4} \cdot \frac{5}{12}} = \frac{14/12}{33/48} = \frac{56}{33}$ .

✓ Check

α + β \approx 0.6435

and

α - β \approx 0.3948

give

2 α \approx 1.0383

, and

t an 1.0383 \approx 1.697 = \frac{56}{33}

⚠ Common trap

(D)

\frac{33}{56}

is the reciprocal, i.e.

co t 2 α

, from inverting the final fraction.

Quick tip

$tan 2 α = tan [(α + β) + (α - β)]$ with $\frac{3}{4}$ and $\frac{5}{12}$

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Trigonometric Functions & Identities — Preparation

This question tests your understanding of Trigonometric Functions & Identities, 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.

Trigonometric functions (sine, cosine, tangent and their reciprocals) relate the angles of a right triangle to the ratios of its sides, and extend to any angle using the unit circle. This topic covers degree and radian measure, values of trigonometric ratios at standard angles, signs in the four quadrants, and allied angles such as (90 - theta) and (180 + theta). It also covers the main identities: sin^2 + cos^2 = 1 and its variations, compound angle formulas such as sin(A + B), multiple and sub-multiple angle formulas such as sin 2A and cos 2A, and the sum-to-product and product-to-sum formulas. Most questions are about choosing the right identity to simplify an expression or prove an equality, so learning these formulas thoroughly pays off. Practise the trigonometric functions and identities questions below, each with a step-by-step solution.

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Trigonometric Functions & Identities — Preparation

If $cos (α + β) = \frac{4}{5}$ and $s in (α - β) = \frac{5}{13}$ , where $0 < α, β < \frac{π}{4}$ , then $t an 2 α$ is equal to