Let `45^(@)lethetalt90^(@) . iftan theta+cottheta=(tan theta)^("i")+(cottheta)^("i")` for some `ige2`, then what is the value of `sin theta+costheta` ?

Options

A. `sqrt(2)`
B. `(1)/(sqrt(2))`
C. `((sqrt(3)+1))/(2)`
D. `(2)/((sqrt(3)+1))`

Answer

A

Explanation

As given , `tantheta+cottheta=(tantheta)^( "i")+(cottheta)^( "i")`
Also , `45^(@)lethetalt90^(@)andi ge2`
which is possible only when `theta=45^(@)`
Since , `tan45^(@)+cot45^(@)=1+1=2`
and `(tan45^(@))^( "i")=1+1=2`
Thus , `sintheta+costheta=sin45^(@)+cos45^(@)`
`=(1)/(sqrt(2))+(1)/(sqrt(2))=(1+1)/(sqrt(2))=(2)/(sqrt(2))=sqrt(2)`