Prove that cos 5 A -16 cos 5 A -20 cos 3 A -5 cos A

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Standard XII

Mathematics

Multiplication of Matrices

Prove that co...

Question

Prove that cos5A= 16cos⁵A -20cos³A +5cosA

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Solution

cos5A = cos (2A + 3A )
= cos2A.cos3A - sin2A.sin3A
= (2cos^2 A -1)(4cos^3-3cosA)-2sinAcosA(3sinA-4sin^3A)
=8cos^5A-6cos^3A - 4cos^3A + 3cosA - 2sin^2AcosA (3-4sin^2A)
= 8cos^5A-6cos^3A - 4cos^3A + 3cosA - 2(1-cos ^2A)cosA(3-4(1-cos^2))
= 8cos^5A-6cos^3A - 4cos^3A + 3cosA - (2cosA-2cos^3A)(4cos^2A-1)
=8cos^5A-6cos^3A - 4cos^3A + 3cosA - 8cos^3A + 8cos^5A+2cosA-2cos^3A
= 16 cos ^5A -20 cos ^3 + 5 cos A

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(xi) $\frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ + ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ + ϕ)}$

Q. If A lies in

2^{n d}

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3 tan A + 4 = 0

the prove that the value of

2 cot A - 5 cos A + sin A

\frac{23}{10}

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Prove that cos5A= 16cos5A -20cos3A +5cosA

Prove that cos5A= 16cos⁵A -20cos³A +5cosA