Verify the Identity (sec(x))/(cos(x))-(tan(x))/(cot(x))=1

Problem

sec(x)/cos(x)−tan(x)/cot(x)=1

Solution

1. Rewrite the first term using the reciprocal identity sec(x)=1/cos(x)

sec(x)/cos(x)=1/cos(x)/cos(x)

2. Simplify the first term by multiplying the numerator by the reciprocal of the denominator.

1/cos(x)⋅1/cos(x)=1/cos2(x)

3. Rewrite the second term using the reciprocal identity cot(x)=1/tan(x)

tan(x)/cot(x)=tan(x)/1/tan(x)

4. Simplify the second term by multiplying the numerator by the reciprocal of the denominator.

tan(x)⋅tan(x)=tan2(x)

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

1/cos2(x)−tan2(x)

6. Apply the reciprocal identity 1/cos2(x)=sec2(x)

sec2(x)−tan2(x)

7. Use the Pythagorean identity 1+tan2(x)=sec2(x) which rearranges to sec2(x)−tan2(x)=1

1=1

Final Answer

sec(x)/cos(x)−tan(x)/cot(x)=1

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