If \(\rm \tan \frac{A}{2}=x\), then find x.

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 18 Jul 2023 Shift 1)

Answer 1

Given:

tan (A/2) = x

Formula used:

tan (A/2) = ± \(\rm \frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}\)

Calculation:

tan (A/2) = x

⇒ \(\rm \frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}\) = x

⇒ x = \(\rm \frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}\)

∴ The correct option is 3.
Alternate Method

We know: tan(A/2) = sin(A/2) / cos(A/2).

Sin(A/2) can be derived by the Pythagorean Identity, sin²λ + cos²λ = 1, which means sinλ = √(1 - cos²λ). So, sin(A/2) = √(1 - cos²(A/2)).

Putting this into the equation, we get

tan(A/2) = √(1 - cos²(A/2)) / cos(A/2) .......(i)

Now, from half angle formula,

cos(A/2) = √((1 + cosA) / 2) .......(ii)

value of (ii) put in equation (i):-

tan(A/2) = √[1 - √{(1 + cosA)²/ 4}] / √((1 + cosA) / 2)

tan(A/2) = √[1 - {(1 + cosA)/ 2}] / √((1 + cosA) / 2)

tan(A/2) = √((2 - 1 - cosA)/ 2) / √((1 + cosA) / 2)

tan(A/2) = √((1 - cosA)/ 2) / √((1 + cosA) / 2)

tan(A/2) = ±√(1 - cosA) / √(1 + cosA).

So, option 3 is the correct answer.

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