Question

If Y = tan^-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`

Answer 1

Let y = tan^-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]`

y = tan^-1 `[((cos 2x) /(cos 2x)- (sin2x)/(cos 2x))/((sin2x) /(cos 2x)- (cos 2x)/(cos 2x))]`

y = tan^-1 `((1 - tan 2x)/(1 + tan 2x))`

y = tan^-1  `[(tan π/4 - tan 2x)/(1 + tan π/4. tan 2x)]`

y = tan^-1 `[tan ( π/4 - 2x)]`

y = π/4 - 2x

Diff w.r.t.x

`therefore (dy)/(dx) = -2`