prove that: quickly
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To prove --->
Sin4A / SinA = 8 Cos³ A - 4 Cos A
Proof --->
LHS = Sin 4A / SinA
We have a formula
Sin 2θ = 2 Sin θ Cos θ
Applying it here
= 2 Sin 2 A Cos 2 A / SinA
Applying above formula again
= 2 ( 2 SinA CosA ) Cos2A / SinA
SinA cancel out from numerator and denominator
= 4 Cos A Cos 2A
We have a formula
Cos 2 A = 2 Cos² A - 1
Applying it here
= 4 CosA ( 2 Cos²A - 1 )
= 4 CosA ( 2 Cos²A ) - 4 CosA ( 1 )
= 8 Cos³ A - 4 CosA = RHS
Additional information--->
1) Cos 2A = Cos² A - Sin² A
2) Cos 2A = 2 Cos²A - 1
3) tan 2A = 2 tanA / (1 - tan²A)
4) Sin 2A = 2 tanA / (1 + tan² A)
5)Cos 2A = (1 - tan²A ) / ( 1 + tan²A )
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