1.

If `cosalpha+cosbeta=(3)/(2)and"sin"alpha+sinbeta=(1)/(2)andtheta` is the arithmetic mean of `alphaandbeta` , then `sin2theta+cos2theta` is equal toA. `(3)/(5)`B. `(7)/(5)`C. `(4)/(5)`D. `(8)/(5)`

Answer» Correct Answer - 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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