prove that tan 90- tan 27° - cot 27° + cot 9° = 4.
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Answer:
LHS = tan9° - tan27° - cot27° + cot9°
= (tan9° + cot9°) - (tan27° + cot27°)
= (sin9°/cos9° + cos9°/sin9°) - (sin27°/cos27° + cos27°/sin27°)
= (sin²9° + cos²9°)/sin9°cos9° - (sin²27° + cos²27°)/sin27° cos27°
= 1/sin9°cos9° - 1/sin27°cos27°
= 2/(2sin9°cos9°) - 2/(2sin27°cos27°)
= 2/sin18° - 2/sin54° [ as we know , sin2x = 2sinxcosx]
= 2 [ (sin54° - sin18°)/sin18°.sin54°]
use formula, sinX - sinY = 2cos(X + Y)/2sin(X - Y)/2
= 2[(2cos36°sin18°)/sin18°sin54°]
= 4[ cos36°/cos(90° - 36°) ]
= 4[ cos36°/cos36° ]
= 4 = RHS
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Answer:
cos 30 sin 50 + 70 = 120 -- 85 = 120
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