Question

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

Answer 1

`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  )`