Question

Prove that: cos(3π/4 + x) - cos(3π/4 - x) = -√2 sin x​

Answer 1

Step-by-step explanation:

Consider

⇛cos{(3π/4) + x} - cos{(3π/4) - x}

We know that

cos x - cos y = 2sin{(x+y)/2}sin{(x-y)/2}

By, using this identity, we get

= -2sin[{(3π/4) + x + (3π/4) - x}/2] sin [{(3π/4) + x - (3π/4) + x}/2]

= -2sin[{(3π/4) + (3π/4)}/2] sin {(x + x)/2}

= -2sin[{2*(3π/4)}/2] sin (2x/2)

= -2sin(3π/4) sinx

= -2sin{x-(π/3)} sinx

= -2sin (π/4)sinx

= -2 * (1/√2) * sinx

= -√(2) * √(2) * (1/√2) * sinx

= -√(2) sinx

= RHS.

Hence, the value of cos{(3π/4) + x} - cos{(3π/4) - x} = -√(2) sinx .

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