Question

Solve cot(x/2) – cosec (x/2) = cotx.​

Answer 1

Question :- Solve cot(x/2) – cosec (x/2) = cotx.

Solution :-

→ cot(x/2) – cosec (x/2) = cotx.

using :-

• cotA = (cosA/sinA)
• cosecA = (1/sinA)

→ cos(x/2)/sin(x/2) - 1/sin(x/2) = cosx /sinx

taking LCM ,

→ [ cos(x/2) - 1 ] / sin(x/2) = cos x / sin x

Now using :-

• sinA = 2 * sin(A/2) * cos(A/2) in denominator,

→ [ cos(x/2) - 1 ] / sin(x/2) = cos x / (2*sin(x/2) * cos(x/2))

→ [ cos(x/2) - 1 ] = cos x / 2 * cos(x/2)

Cross - Multiply,

→ 2cos²(x/2) - 2 cos(x/2) = cos x

Now, using :-

• cos A = 2cos²(A/2) - 1 in denominator,

→ 2cos²(x/2) - 2 cos(x/2) = 2cos²(x/2) - 1

→ 2 cos(x/2) = 1

→ cos(x/2) = 1/2

→ cos(x/2) = cos(π/3)

→ x/2= 2npi ± (π/3)

→ x = 4npi ± (2π/3). { n € Z. }

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