If `cos 5 theta = a cos^(5)theta + b cos^(3)theta+c cos theta`. Then, find the value of c.

Options

A. `-5`
B. 1
C. 5
D. none of these

Answer

C

Explanation

We have,
`cos 5 theta+i sin5 theta=(cos theta+i sintheta)^(5)`
`=cos5 theta+isin5 theta`
`={""^(5)C_(0)0-""^(5)C_(2)cos^(3)thetasin^(2)theta+""^(5)C_(4)cos thetasin^(4)theta}`
`" "+i{""^(5)C_(1)cos^(4)thetasintheta-""^(5)C_(3)cos^(2)thetasin^(3)theta....}`
`impliescos 5 theta=cos^(5)theta-10cos^(3)thetasin^(2)theta+5 costhetasin^(4)theta`
`impliescos5 theta=cos^(5)theta-10cos^(3)theta(1-cos^(2)theta)+5costheta(1-cos^(2)theta)^(2)`
`impliescos5theta=16cos^(5)theta-20cos^(3)theta+5 cos theta`
`thereforec=5`
Differentiating with respect to `theta,` we get
`-5 sin5 theta=-5a cos^(4)thetasintheta-3bcos^(2)thetasintheta-csintheta`
Putting `theta=(pi)/(2),` we get `c=5.`