3.3: Solving Trig Equations using Identities

Answer 1

cot(−x)cos(−x)+sin(−x)=−1/f(x)

By negative angle identity:

-cot(x)cos(x)-sin(x) = -1/f(x)

By quotient identity for cot (x)

-[cos(x)/sin(x)] cos(x) -sin(x) = -1/f(x)

-cos²(x)/ sin(x) - sin(x) = -1/f(x)

[-cos²(x)- sin²(x)]/sin(x)= -1/f(x)

factor out negative one on left side:

-[cos²(x)+sin²(x)]/sin(x) = -1/f(x)

Pythagorean Identity: cos²(x)+sin²(x) =1

-1/sin(x) = -1/f(x)

Get the reciprocal of both sides:

-sin(x) =-f(x)

Therefore:

f(x) = sin(x)