Question

Cos^2x + cos^2( x + 120) + cos^2( x - 120)

Answer 1

To Find :-

• value of cos²x + cos²(x + 120) + cos²(x - 120) = ?

Formula used :-

→ Cos2A = 2Cos²A - 1

→ Cos2A + 1 = 2Cos²A

→ (1 + Cos2A)/2 = Cos²A

→ Cos²A = (1 + Cos2A)/2

Solution :-

with Above Formula we can written :-

→ cos²(x + 120) = {1 + cos2(x + 120)/2 } = {1 + cos(2x + 240)/2}

→ cos²(x - 120) = {1 + cos2(x - 120) /2 } = {1 + cos(2x - 240)/2}

Adding These Two values we get :-

→ {1 + cos(2x + 240)/2} + {1 + cos(2x - 240)/2}

→ [{ 1 + cos(2x + 240) } + {1 + cos(2x - 240) } ] / 2

→ [ 2 + cos(2x + 240) + cos(2x - 240) ] /2

→ 1 + [ cos(2x + 240) + cos(2x - 240) ] /2

Using cosA + cosB = 2 * cos(A + B/2) * cos(A - B/2)

→ 1 + [2 * cos{(2x + 240 + 2x - 240)/2} * cos{(2x + 240 - 2x + 240)/2} ] /2

→ 1 + [2 * cos(4x/2) * cos(480/2) ] /2

→ 1 + [ 2 * cos2x * cos240 ] /2

→ 1 + (cos2x * cos240° ]

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Now , Putting This value in Our Question we get,

→ cos²x + 1 + (cos2x * cos240°)

Now,

→ cos240° = cos(180° + 60°) = -(cos60° = -(1/2) .

→ cos²x + 1 + cos2x * (-1/2)

Putting cos²x = (1 + cos2x)/2 we get,

→ 1 + {(1 + cos2x)/2} - (cos2x/2)

→[2 + 1 + cos2x - cos2x ] /2

→ (3/2) (Ans).

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