Simplify 1/(cot(x)^2)+sec(x)cos(x)

Problem

1/cot2(x)+sec(x)*cos(x)

Solution

1. Apply reciprocal identities to the first term by using the fact that 1/cot(x)=tan(x)

1/cot2(x)=tan2(x)

2. Apply reciprocal identities to the second term by using the fact that sec(x)=1/cos(x)

sec(x)*cos(x)=1/cos(x)⋅cos(x)

3. Simplify the second term by canceling the common factor of cos(x)

1/cos(x)⋅cos(x)=1

4. Substitute the simplified terms back into the original expression.

tan2(x)+1

5. Apply the Pythagorean identity which states that tan2(x)+1=sec2(x)

tan2(x)+1=sec2(x)

Final Answer

1/cot2(x)+sec(x)*cos(x)=sec2(x)

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