Grade 11Trigonometry
Question Prove that : ...Cos5a =16cos^5a - 20cos^3a + 5cosa
1 Answer
Vikas TU
Consider LHS
we can writ it as:
cos(5A) = cos(2A+3A)
= cos(2A)cos(3A) - sin(2A)sin(3A)1
we realize 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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