If `(sin(theta+alpha))/(cos(theta-alpha))=(1-m)/(1+m)`, prove that `tan(pi/4-theta)tan(pi/4-alpha)=m`

If `(sin(theta+alpha))/(cos(theta-alpha))=(1-m)/(1+m)`, prove that `ta

1.	If `(sin(theta+alpha))/(cos(theta-alpha))=(1-m)/(1+m)`, prove that `tan(pi/4-theta)tan(pi/4-alpha)=m`
Answer» Let, `pi/4 - theta = A and pi/4-alpha = B` Then`pi/4-theta+pi/4-alpha = A+B` `=>theta+alpha = pi/2-(A+B)` `=>sin(theta+alpha) = sin(pi/2-(A+B))` `=>sin(theta+alpha) = cos(A+B)` Similarly, `cos(theta-alpha) = cos(B-A)` Now, putting these values in the given equation, `(cos(A+B))/(cos(B-A)) = (1-m)/(1+m)` `=>(cos(B-A))/(cos(A+B)) = (1+m)/(1-m)` Using componendo and dividendo, `=>(cos(B-A)+cos(A+B))/(cos(B-A)-cos(A+B)) = (1+m+1-m)/(1+m-1+m)` `=>(2cosAcosB)/(2sinAsinB) = 2/(2m)` `=>1/(tanAtanB) = 1/m` `=>tanAtanb = m` `:. tan(pi/4-theta)tan(pi/4-alpha) = m.`