If `sec theta+tan theta=2+sqrt5`, then the value of `sin_(theta)=cos_(theta)` is: (यदि `sec theta+tan theta=2+sqrt5` है तो का मान है )

Options

A. `(3)/(sqrt5)`
B. `sqrt5`
C. `(7)/(sqt5)`
D. `(1)/(sqrt5)`

Answer

a

Explanation

`sec theta+tan theta=2+sqrt(5)`
`sec^(2)theta-tan^(2)theta=1`
`(sec theta-tantheta)(sectheta+tan theta)=1`
`(Sec theta-tantheta)=(1)/(2+sqrt(5))=(1)/(sqrt(5+2))`
`sqrt(5)-2`
adding eq (i) +(ii)
`2 sec theta=2+sqrt(5)+sqrt(5)-2`
`rArr 2sectheta=2sqrt( 5)`
`rArr sectheta=sqrt(5)`
`rArr cos theta=(1)/sqrt(5)`
`rArrsin^(2)theta=1-((1)/sqrt(5))^(2)`
`rArrsin^(2)theta=(4)/(5)`
`rArr sintheta=(2)/sqrt(5)`
`therefore sin theta+cos theta+(2)/sqrt(5)+(1)/sqrt(5)=(3)/sqrt(5)`