Simplify (csc(x)cot(x))/(sec(x))

Problem

(csc(x)*cot(x))/sec(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the fundamental identities csc(x)=1/sin(x) cot(x)=cos(x)/sin(x) and sec(x)=1/cos(x)

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

2. Multiply the terms in the numerator.

cos(x)/sin2(x)/1/cos(x)

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

cos(x)/sin2(x)⋅cos(x)/1

4. Combine the terms into a single fraction.

cos2(x)/sin2(x)

5. Apply the cotangent identity cot(x)=cos(x)/sin(x) to reach the final simplified form.

(cos(x)/sin(x))2=cot2(x)

Final Answer

(csc(x)*cot(x))/sec(x)=cot2(x)

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