Verified Answer :
\(\bf cos(6\theta)\implies cos(3\theta+3\theta)\implies cos[2(3\theta)]
\\\\\\
now\implies
\begin{cases}
cos(3x)=4cos^3(x)-3cos(x)\\\\
cos(2x)=2cos^2(x)-1
\end{cases}
\\\\\\
cos[2(3\theta)]\implies 2\left[ 4cos^3(\theta)-3cos(\theta) \right]^2-1\)
so.. just expand the binomial, and simplify if any
\(\bf sin(6\theta)\implies sin(3\theta+3\theta)\implies sin[2(3\theta)]
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now\implies
\begin{cases}
sin(2x)=2cos(x)sin(x)\\\\
sin(3x)=3sin(x)-4sin^3(x)
\end{cases}
\\\\\\
sin[2(3\theta)]\implies 2cos(3\theta)sin(3\theta)
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2\ [4cos^3(\theta)-3cos(\theta)]\ [3sin(\theta)-4sin^3(\theta)]\)
multiply and distribute and simplify if any
The expansion for \(\rm cos\; 6\theta\) and \(\rm sin\; 6\theta\) is given below and this can be evaluated by using the trigonometric properties.
Given :
Trigonometric Functions -- \(\rm cos\; 6\theta\) and \(\rm sin\; 6\theta\)
The following steps can be used in order to determine the expansion for \(\rm cos\; 6\theta\) and \(\rm sin\; 6\theta\):
Step 1 - The trigonometric properties can be used in order to determine the expansion for \(\rm cos\; 6\theta\) and \(\rm sin\; 6\theta\).
Step 2 - The properties are used is mentioned below:
\(\rm cos\;2\theta = 2cos^2\theta - 1\)
\(\rm cos\; 3\theta=4cos^3\theta - 3cos\theta\)
\(\rm sin\;2\theta = 2\;sin\;\theta \;cos\;\theta\)
\(\rm sin\; 3\theta=3sin\theta-4sin^3\theta\)
Step 3 - The expansion of \(\rm cos\; 6\theta\) is evaluated below:
\(\rm cos\; 6\theta = cos 2(3\theta)\)
\(\rm cos\; 6\theta = 2(4cos^3\theta-3cos\theta)^2-1\)
\(\rm cos\; 6\theta = 2(16cos^6\theta+9cos^2\theta-24cos^4\theta)-1\)
\(\rm cos\; 6\theta = 32cos^6\theta+18cos^2\theta-48cos^4\theta-1\)
Step 4 - The expansion of \(\rm sin\; 6\theta\) is evaluated below:
\(\rm sin\; 6\theta = sin2(3\theta)\)
\(\rm sin\; 6\theta = 2\;sin3\theta \; cos3\theta\)
\(\rm sin\; 6\theta = 2(4cos^3\theta-3cos\theta)(3sin\theta -4sin^3\theta)\)
\(\rm sin\; 6\theta = 2(12cos^3\theta sin\theta-16cos^3\theta sin^3\theta-9cos\theta sin\theta+12cos\theta sin^3\theta)\)
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https://brainly.com/question/13710437