Verify the Identity 1/(sec(x)tan(x))=csc(x)-sin(x)

Problem

1/(sec(x)*tan(x))=csc(x)−sin(x)

Solution

1. Rewrite the left side using the reciprocal and quotient identities for sec(x) and tan(x)

1/(sec(x)*tan(x))=1/(1/cos(x)⋅sin(x)/cos(x))

2. Simplify the denominator by multiplying the fractions.

1/sin(x)/cos2(x)

3. Invert the fraction to simplify the expression.

cos2(x)/sin(x)

4. Apply the Pythagorean identity cos2(x)=1−sin2(x) to the numerator.

(1−sin2(x))/sin(x)

5. Split the fraction into two separate terms.

1/sin(x)−sin2(x)/sin(x)

6. Simplify each term using the reciprocal identity csc(x)=1/sin(x) and basic division.

csc(x)−sin(x)

Final Answer

1/(sec(x)*tan(x))=csc(x)−sin(x)

Want more problems? Check here!