Question

The value of sin 12 theta/sin 4 theta - cos 12 theta/cos 4 theta is?

Answer 1

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Answer 2

Answer:

The value of (sin12θ/sin4θ) - (cos12θ/cos4θ) = 2.

Step-by-step explanation:

List of trigonometric formulas

• We only take into account trigonometric formulas for right-angled triangles when we first learn about them.
• Three sides make up a right-angled triangle: the hypotenuse, the opposite side (perpendicular), and the adjacent side (Base).
• The side with the greatest length is called the hypotenuse, the side perpendicular to the angle is called the opposite side, and the adjacent side is the side on which both the hypotenuse and the opposite side rest.

Sum and difference Identities:

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• sin(x-y) = sin(x)cos(y)-cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)-sin(x)sin(y)
• cos(x-y) = cos(x)cos(y)+sin(x)sin(y)
• tan(x+y) = (tanx+tany)/(1-tanxtany)

Double Angle Identities:

• sin(2x) = 2sinxcosx
• cos2x = cos²x-sin²x = 2cos²x-1 = 1-2sin²x
• tan2x = 2tanx/1-tan²x

Given trigonometric expression as

    (sin12θ/sin4θ) - (cos12θ/cos4θ)

  = (sin12θcos4θ-sin4θcos12θ)/(sin4θcos4θ)

  = sin(12θ-4θ)/(sin4θcos4θ)         [sin(x-y) = sinxcosy-cosxsiny]

  = sin8θ/(sin4θcos4θ)

  now multiply and divide with 2 in the denominator

  = sin8θ/[(2sin4θcos4θ)/2]

  = 2sin8θ/2sin4θcos4θ

  = 2sin8θ/sin2(4θ)                     [sin2x = 2sinxcosx]

  = 2sin8θ/sin8θ

  = 2

Hence, the value of (sin12θ/sin4θ) - (cos12θ/cos4θ) = 2.

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