Cot alpha + cot (60 + Alpha) -Cot (60 -alpha)=3cot alpha
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cota+cot(60+a)−cot(60−a)=cota+cos(60+a)sin(60+a)−cos(60−a)sin(60−a)=cota+cos(60+a)sin(60−a)−cos(60−a)sin(60+a)sin(60+a)sin(60−a)=cot a+sin{(60−a)−(60+a)sin(60+a)sin(60−a)=cota −sin2asin260−sin2a=cosasina−sin2a34−sin2a=(34−sin2a)cosa−sin2a(sina)sina(34−sin2a)=(3−4sin2a)cosa−2*4sinacosa sinasina(3−4sin2a)=3cosa−4sin2acosa−8sin2acosa3sina−4sin3a=3cosa−12sin2acosa3sina−4sin3a=3cosa−12(1−cos2a)cosa3sina−4sin3a=12cos3a−9cosa3sina−4sin3a=3(4cos3a−3cosa)3sina−4sin3a=3cos3asin3a=3cot3a
cota+cot(60+a)−cot(60−a)=cota+cos(60+a)sin(60+a)−cos(60−a)sin(60−a)=cota+cos(60+a)sin(60−a)−cos(60−a)sin(60+a)sin(60+a)sin(60−a)=cot a+sin{(60−a)−(60+a)sin(60+a)sin(60−a)=cota −sin2asin260−sin2a=cosasina−sin2a34−sin2a=(34−sin2a)cosa−sin2a(sina)sina(34−sin2a)=(3−4sin2a)cosa−2*4sinacosa sinasina(3−4sin2a)=3cosa−4sin2acosa−8sin2acosa3sina−4sin3a=3cosa−12sin2acosa3sina−4sin3a=3cosa−12(1−cos2a)cosa3sina−4sin3a=12cos3a−9cosa3sina−4sin3a=3(4cos3a−3cosa)3sina−4sin3a=3cos3asin3a=3cot3a
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L.H.S. =
sinα
cos
sin(60
∘
+α)
cos(60
∘
+α)
−
sin(60
∘
−α)
cos(60
∘
−α)
sin(60
∘
−α)cos(60
∘
+α)
=
sinα
cosα
+
sin
2
−60
∘
−sin
2
α
−cos(60
∘
−α)sin(60
∘
+α)
=
sinα(
4
3
−sin
2
α)
cosα(
4
3
−sin
2
α)+sinαsin(60
∘
−α−60
∘
−α)
=
4sinα(
4
3−4sin
2
α
)
cosα(3−4sin
2
α)+4sinαsin(−2α)
=
3sinα−4sin
3
α
cosalpha(3−4sin
2
α)−4sinα⋅sinαcosα
=
sin3α
cosα[3−12(1−cos
2
α)]
=3⋅
sin3alpha
[4cos
3
α−3cosalpha]
=3⋅
sin3alpha
cos3alpha
=3cot3α
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