Simplifying trigonometry

Question

Simplify the trigonometric expression below by writing the simplified form in terms of cos(x).

(tan(x)+cot(x))/csc(x)

Answer 1

Recall that tan(x) = sin(x)/cos(x) and cot(x) = 1/tan(x) = cos(x)/sin(x).

Step one: Express tan(x)+cot(x) as one fraction in terms of cos(x) and sin(x);

tan(x)+cot(x)=sin(x)/cos(x) + cos(x)/sin(x) = (sin^2(x)+cos^2(x))/(sin(x)cos(x)) = 1/(sin(x)cos(x))

Step two: Multiply the above with 1/csc(x) = 1/(1/sin(x)) = sin(x)

(tan(x)+cot(x)) * 1/csc(x) = 1/(sin(x)cos(x)) * sin(x) = 1/cos(x) = sec(x)