Question

(cos40*+isin40*)^2(cos20*+isin20*)^4(cos10*-isin10*)^4

Answer 1

Answer:

LHS = (cos30° - sin20°)/(cos40° + cos20°)

We know, sin(90-Ф) = cosФ

so, sin20° = sin(90°-70°) = cos70°

∴ (cos30° - cos70°)/(cos40° + cos20°)

Use formula, cosC - cosD = 2sin(C + D)/2.sin(D - C)/2

cosC + cosD = 2cos(C + D)/2.cos(C - D)/2

so, (cos30° - cos70°)/(cos40° + cos20°) = 2sin50°sin20°/2cos30°.cos10°

We also know, sin2Ф = 2sinФ.cosФ

so, sin20° = 2sin10°.cos10°, put it in above

= 4sin50°.sin10°.cos10°/2(√3/2)cos10° [∵ cos30° = √3/2 ]

= 4sin(90-40°)sin10°/√3

=4/√3 cos40° sin(90°-80°)

= 4/√3 cos40°cos80° = RHS

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