cos6 A + sin6 A = 1 - 3sin2 A . cos2A
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EXPLANATION.
⇒ cos⁶(A) + sin⁶(A) = 1 - 3sin²A.cos²A.
As we know that,
We can write equation as,
⇒ [(cos²A)³ + (Sin²A)³].
As we know that,
Formula of :
⇒ a³ + b³ = (a + b)³ - 3ab(a + b).
Using this formula in equation, we get.
⇒ [cos²A + sin²A]³ - 3cos²A.sin²A(cos²A + sin²A).
As we know that,
Formula of :
⇒ sin²x + cos²x = 1.
Using this formula in equation, we get.
⇒ 1 - 3cos²A.sin²A.
Hence proved.
MORE INFORMATION.
Trigonometrical ratios of multiple angles.
(1) = sin2θ = 2sinθcosθ = 2tanθ/1 + tan²θ.
(2) = cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ = 1 - tan²θ/1 + tan²θ.
(3) = tan2θ = 2tanθ/1 - tan²θ.
(4) = sin3θ = 3sinθ - 4sin³θ.
(5) = cos3θ = 4cos³θ - 3cosθ.
(6) = tan3θ = 3tanθ - tan³θ/1 - 3tan²θ.
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