Simplify: `cos^(-1)((sinx+cosx)/(sqrt(2))),\ \ pi/4

Answer

N/a

Explanation

Let `y = cos^(-1)((sinx+cosx)/(sqrt(2)))`
` :. (dy)/(dx)=d/dx cos^(-1)((sinx+cosx)/(sqrt(2)))`
`= (-1)/(sqrt(1-((sinx+cosx)/sqrt(2))^(2))).(d)/(dx)((sinx+cosx)/(sqrt(2)))`
`[:' (d)/(dx)(cosx)= -(1)/(sqrt(1-x^(2)))]`
`= (1)/(sqrt(4-(sin^(2)x+cos^(2)x+2sinx.cosx)/(2))).(1)/(sqrt(2))(cosx-sinx)`
` =(-1.sqrt(2))/(sqrt(1-sin2x)).(1)/(sqrt(2))(cosx-sinx)`
` [:'1-sin2x=(cosxsinx)^(2)=cos^(2)x+sin^(2)x-2sinxcosx]`
`= (-1(cosx-sinx))/((cosx-sinx)) = - 1`