If _i=1^ntanα _i=1, where each α _i∈[0,π /2), then the maximum value of _i=1^nsinα _i is

B. 12^n/2. Step 1: ∏tanα _i=1 means ∏sinα _i=∏cosα _i. Step 2: So (∏sinα _i)^2=∏sinα _icosα _i=12^n∏sin2α _i≤ 12^n. Step 3: Hence ∏sinα _i≤ 12^n/2, with equality when every α _i=π /4. Check: n=2, α _1=α _2=π /4: tanα _1tanα _2=1 and sinα _1sinα _2=1/2=12^2/2. Trap: (A) 12^n is the bound for the square (∏sinα _i)^2; taking the square root gives 12^n/2.

Question 8 of 47 · Trigonometric Functions & Identities

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NIMCET 2023

If $\prod_{i = 1}^{n} t an α_{i} = 1$ , where each $α_{i} \in [0, \frac{π}{2})$ , then the maximum value of $\prod_{i = 1}^{n} s in α_{i}$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $\frac{1}{2 ^{n /2}}$

Detailed Solution

$\prod t an α_{i} = 1$ means $\prod s in α_{i} = \prod cos α_{i}$ .

So $(\prod s in α_{i})^{2} = \prod s in α_{i} cos α_{i} = \frac{1}{2 ^{n}} \prod s in 2 α_{i} \leq \frac{1}{2 ^{n}}$ .

Hence $\prod s in α_{i} \leq \frac{1}{2 ^{n /2}}$ , with equality when every $α_{i} = \frac{π}{4}$ .

✓ Check

n = 2

α_{1} = α_{2} = \frac{π}{4}

t an α_{1} t an α_{2} = 1

and

s in α_{1} s in α_{2} = \frac{1}{2} = \frac{1}{2 ^{2/2}}

⚠ Common trap

(A)

\frac{1}{2 ^{n}}

is the bound for the square

(\prod s in α_{i})^{2}

; taking the square root gives

\frac{1}{2 ^{n /2}}

Quick tip

$(\prod sin α_{i})^{2} = \prod sin α_{i} cos α_{i} = \frac{1}{2 ^{n}} \prod sin 2 α_{i} \leq \frac{1}{2 ^{n}}$

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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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If ∏i=1n​tanαi​=1, where each αi​∈[0,2π​), then the maximum value of ∏i=1n​sinαi​ is

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Trigonometric Functions & Identities — Preparation

If $\prod_{i = 1}^{n} t an α_{i} = 1$ , where each $α_{i} \in [0, \frac{π}{2})$ , then the maximum value of $\prod_{i = 1}^{n} s in α_{i}$ is