Question

derivative of x+sin x divided by x-sin x

Answer 1

Given:  sin(x) = cos(x); Chain Rule.
Solve:

cos(x) = sin(x + PI/2)

 cos(x) =  sin(x + PI/2)

=  sin(u) *  (x + PI/2) (Set u = x + PI/2)

= cos(u) * 1 = cos(x + PI/2) = -sin(x)       Q.E.D.

Proof of  tan(x) : from the derivatives of sine and cosine

Given:  sin(x) = cos(x);  cos(x) = -sin(x); Quotient Rule.
Solve:

tan(x) = sin(x) / cos(x)

 tan(x) =  sin(x)/cos(x)

= ( cos(x)  sin(x) - sin(x)  cos(x) ) / cos^2(x)

= ( cos(x)cos(x) + sin(x)sin(x) ) / cos^2(x)

= 1 + tan^2(x) = sec^2(x)       Q.E.D.