Question

Prove that cos^5A=16cos^5A-20cos^3A+5cosA

Answer 1

Step-by-step explanation:

Consider LHS

we can writ it as:

cos(5A) = cos(2A+3A)

= cos(2A)cos(3A) - sin(2A)sin(3A)1

let consider that Sin 3A= 3 sin(A)cos²(A) − sin³(A)

Sin 2A= 2sinAcosA

Cos 3A=cos³(A) − 3 sin²(A)cos(A)

Cos 2A=cos²A−sin²A

Sub in eq.1

Cos 5A=.cos⁵A − 3sin²Acos³A − sin²Acos³A + 3sin⁴AcosA − 6sin²Acos³A + 2sin⁴AcosA

= 5sin⁴AcosA − 10sin²Acos³A + cos⁵A

= 5(1−cos²A)²cosA − 10(1−cos²A)cos³A + cos⁵A

= 5cosA(cos⁴A−2cos²A+1) − 10cos³A(1−cos²A) + cos⁵A

= 5cos⁵A − 10cos³A + 5cosA − 10cos³A + 10cos⁵A + cos⁵A

= 16cos⁵A − 20cos³A + 5cosA

=RHS

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