The sum of all possible solution of trigonometric equation `2sqrt(cos^(6)theta-sin^(6)theta)=cos2theta+1` in interval `[pi,7pi]` is equal to

Options

A. `7pi`
B. `14pi`
C. `28pi`
D. `42pi`

Answer

A

Explanation

Given, `2sqrt(cos^(6)theta-sin^(6)theta)=cos2theta+1`
`4(cos^(6)theta-sin^(6)theta)=(cos2theta+1)^(2)implies4((1+cos2theta)/(2))^(3)-4((1-cos2theta)/(2))^(3)=(1+cos2theta)^(2)`
`implies" "(1+cos2theta)^(3)-(1-cos2theta)^(3)=2[1+2cos2theta+cos^(2)2theta]`
`implies" "2(cos^(3)2theta+3cos2theta)=(1+2cos2theta+cos^(2)2theta)`
`implies" "cos^(3)2theta-cos^(2)2theta+cos2theta-1=0`
`implies" "(cos2theta-1)(cos^(2)2theta+1)=0`
`:." "cos2theta=1" "implies" "2theta=2npi`
`" "theta=npi," "ninI`
Sum of al solution `=pi+2pi+3pi+.........+7pi`
`=pi(1+2+3+.........+7)`
`=pi((7)(8))/(2)=28pi`