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If u=(1+cos theta)(1+cos 2theta)-sin the...

If `u=(1+cos theta)(1+cos 2theta)-sin theta.sin 2theta, v=sin theta(1+cos 2theta)+sin 2theta(1+cos theta)`, then `u^(2)+v^(2)=`

A

`4(1+cos theta)(1+cos 2theta)`

B

`4(1+sin theta)(1+sin 2theta)`

C

`4(1-cos theta)(1-cos 2theta)`

D

`4(1-sin theta)(1-sin 2theta)`

Text Solution

Verified by Experts

The correct Answer is:
A
`u=1+cos theta + cos 2theta + cos (theta + 2theta)`
`=(1+cos 3theta)+(cos theta + cos 2theta)`
`=2 "cos"^(2)(3theta)/(2)+2 cos.(3theta)/(2)cos.(theta)/(2)`
`=2 cos.(3theta)/(2)[cos.(3theta)/(2)+cos.(theta)/(2)]`
Similarly `v=2 sin.(3theta)/(2)[cos.(3theta)/(2)+cos.(theta)/(2)]`
`therefore u^(2)+v^(2)=4(cos.(3theta)/(2)+cos.(theta)/(2))^(2)`
`=4(2cos theta. cos.(theta)/(2))^(2)=16 cos^(2)theta."cos"^(2)(theta)/(2)`
`= 4(1+cos theta)(1+cos 2 theta)`
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