Simplify csc(x)*sec(x)-tan(x)

Problem

csc(x)*sec(x)−tan(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal and quotient identities.

csc(x)=1/sin(x)

sec(x)=1/cos(x)

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

2. Substitute these identities into the original expression.

1/sin(x)⋅1/cos(x)−sin(x)/cos(x)

3. Combine the first two fractions.

1/(sin(x)*cos(x))−sin(x)/cos(x)

4. Find a common denominator to subtract the fractions. The common denominator is sin(x)*cos(x)

1/(sin(x)*cos(x))−(sin(x)⋅sin(x))/(cos(x)⋅sin(x))

1/(sin(x)*cos(x))−sin2(x)/(sin(x)*cos(x))

5. Subtract the numerators over the common denominator.

(1−sin2(x))/(sin(x)*cos(x))

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

cos2(x)/(sin(x)*cos(x))

7. Cancel the common factor of cos(x) from the numerator and denominator.

cos(x)/sin(x)

8. Identify the resulting expression as the cotangent function.

cot(x)

Final Answer

csc(x)*sec(x)−tan(x)=cot(x)

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