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If cosalpha+cosbeta=(3)/(2)and"sin"alpha...

If `cosalpha+cosbeta=(3)/(2)and"sin"alpha+sinbeta=(1)/(2)andtheta` is the arithmetic mean of `alphaandbeta` , then `sin2theta+cos2theta` is equal to

A

`(3)/(5)`

B

`(7)/(5)`

C

`(4)/(5)`

D

`(8)/(5)`

Text Solution

Verified by Experts

The correct Answer is:
B
`cosalpha+cosbeta=(3)/(2)andsinalpha+sinbeta=(1)/(2)`
`rArr2cos((alpha+beta)/(2))cos((alpha-beta)/(2))=(3)/(2)`
and `2sin((alpha+beta)/(2))cos((alpha-beta)/(2))=(1)/(2)`
`rArrtan((alpha+beta)/(2))=(1)/(3)`
`becausetheta=(alpha+beta)/(2)` [Given]
`rArr2theta=alpha+beta`
`thereforesin2theta+cos2theta=sin(alpha=beta)+cos(alpha+beta)`
`=(2tan((alpha+beta)/(2)))/(1+tan^(2)((alpha+beta)/(2)))+(1-tan^(2)((alpha+beta)/(2)))/(1+tan^(2)((alpha+beta)/(2)))`
`[becausesin2theta=(2tantheta)/(1-tan^(2)theta),cos2theta=(1-tan^(2)theta)/(1+tan^(2)theta)]`
`=(2((1)/(3)))/(1+((1)/(3))^(2))+(1-((1)/(3))^(2))/(1+((1)/(3))^(2))=(2)/(3)xx(9)/(10)+(8)/(9)xx(9)/(10)=(6)/(10)+(8)/(10)=(7)/(5)`
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