The value of sin^23θ sin^2θ -cos^23θ cos^2θ is equal to

A. 8cos2θ. Step 1: sin3θ /sinθ =3-4sin^2θ =1+2cos2θ and cos3θ /cosθ =4cos^2θ -3=2cos2θ -1. Step 2: The expression is a difference of squares: (1+2cos2θ )^2-(2cos2θ -1)^2. Step 3: With a=1+2cos2θ and b=2cos2θ -1: a-b=2 and a+b=4cos2θ, so the value is 2⋅4cos2θ =8cos2θ. Check: At θ =π /6 the expression is 1/1/4-0=4, and 8cosπ /3=4. Trap: Expanding sin3θ and cos3θ separately is long and error-prone; the difference-of-squares route is faster.

Question 4 of 47 · Trigonometric Functions & Identities

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The value of $\frac{s in ^{2} 3 θ}{s in ^{2} θ} - \frac{cos ^{2} 3 θ}{cos ^{2} θ}$ is equal to

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Answer: Option A — $8 cos 2 θ$

Detailed Solution

$\frac{s in 3 θ}{s in θ} = 3 - 4 s in^{2} θ = 1 + 2 cos 2 θ$ and $\frac{cos 3 θ}{cos θ} = 4 cos^{2} θ - 3 = 2 cos 2 θ - 1$ .

The expression is a difference of squares: $(1 + 2 cos 2 θ)^{2} - (2 cos 2 θ - 1)^{2}$ .

With $a = 1 + 2 cos 2 θ$ and $b = 2 cos 2 θ - 1$ : $a - b = 2$ and $a + b = 4 cos 2 θ$ , so the value is $2 \cdot 4 cos 2 θ = 8 cos 2 θ$ .

✓ Check

θ = \frac{π}{6}

the expression is

\frac{1}{1/4} - 0 = 4

, and

8 cos \frac{π}{3} = 4

⚠ Common trap

Expanding

s in 3 θ

and

cos 3 θ

separately is long and error-prone; the difference-of-squares route is faster.

Quick tip

$\frac{s i n 3 θ}{s i n θ} = 1 + 2 cos 2 θ, \frac{c o s 3 θ}{c o s θ} = 2 cos 2 θ - 1$

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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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The value of sin2θsin23θ​−cos2θcos23θ​ is equal to

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Trigonometric Functions & Identities — Preparation

The value of $\frac{s in ^{2} 3 θ}{s in ^{2} θ} - \frac{cos ^{2} 3 θ}{cos ^{2} θ}$ is equal to