Question

If `y = log tan (π/4 + x/2)`, then prove that `(dy)/(dx) - secx = 0`.

Answer 1

`y = log tan (π/4 + x/2)`

Differentiating with respect to x:

`(dy)/(dx) = 1/(tan(π/4 + x/2)) xx sec^2 (π/4 + x/2) xx 1/2`

= `(cos(π/4 + x/2))/(sin(π/4 + x/2)) xx 1/(cos^2(π/4 + x/2)) xx 1/2`

= `1/(2sin(π/4 + x/2) cos(π/4 + x/2))`

= `1/(sin(π/2 + x)`

= `1/cosx`

= sec x

`(dy)/(dx) - sec x = 0`

Hence proved.