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write an equivalent expression for cos^4 theta that does not involve any powers of sin or cosine greater then 1.

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Answer:


cos^4\Theta =(1)/(4)(2+(cos4\Theta )/(2)+2cos\Theta  )

Explanation:

We have given
cos^4\Theta

We have to find the expression in which there is no term of sine or cos of power more than 1

So
cos^4\Theta =cos^2\Theta * cos^2\Theta

We know that
cos2\Theta =2cos^2\Theta -1


cos^2\Theta =(1+cos2\Theta )/(2)

So
cos^4\Theta =(1+cos2\Theta )/(2)* (1+cos2\Theta )/(2)


cos^4\Theta =(1)/(4)(1+cos2\Theta )^2


cos^4\Theta =(1)/(4)(1+cos^22\Theta+2cos\Theta  )


cos^4\Theta =(1)/(4)(1+(1+(cos4\Theta )/(2))+2cos\Theta  )


cos^4\Theta =(1)/(4)(2+(cos4\Theta )/(2)+2cos\Theta  )

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