Prove the following: 6x=32 ^6x-48 ^4x+18 ^2x-1 - Vidyadip

Question

Prove the following: $\cos6\text{x}=32\cos^6\text{x}-48\cos^4\text{x}+18\cos^2\text{x}-1$

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Answer

$\text{L.H.S.}=\cos6\text{x}$ $=\cos3(2\text{x})$ $=4[(2\cos^2\text{x}-1)^3-3(2\cos^2\text{x}-1)[\cos2\text{x}=2\cos^2\text{x}-1]$ $=4[(2\cos^2\text{x})^3-(1)^3-3(2\cos^2\text{x})^2+3(2\cos^2\text{x}=2\cos^2\text{x})]-\cos^2\text{x}+3$ $=4[8\cos^6\text{x}-1-12\cos^4\text{x}+6\cos^2\text{x}]-6\cos^2\text{x}+3$ $=32\cos^6\text{x}-4-48\cos^4\text{x}+24\cos^2\text{x}-6\cos^2\text{x}+3$ $=32\cos^6\text{x}-48\cos^4\text{x}+18\cos^2\text{x}-1=\text{R.H.S.}$

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