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

Problem

tan(x)/(1−cos(x))=csc(x)*(1+sec(x))

Solution

1. Rewrite the left side of the identity using the quotient identity for tangent.

tan(x)/(1−cos(x))=sin(x)/cos(x)/(1−cos(x))

2. Simplify the complex fraction by moving the denominator of the numerator into the main denominator.

sin(x)/(cos(x)*(1−cos(x)))

3. Multiply the numerator and the denominator by the conjugate of the denominator's binomial factor, which is 1+cos(x)

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

4. Expand the product of the conjugates in the denominator using the difference of squares.

(sin(x)*(1+cos(x)))/(cos(x)*(1−cos^2(x)))

5. Substitute the Pythagorean identity sin^2(x)=1−cos^2(x) into the denominator.

(sin(x)*(1+cos(x)))/(cos(x)*sin^2(x))

6. Cancel a factor of sin(x) from the numerator and the denominator.

(1+cos(x))/(cos(x)*sin(x))

7. Split the fraction into two separate terms over the common denominator.

1/(cos(x)*sin(x))+cos(x)/(cos(x)*sin(x))

8. Simplify each term by canceling cos(x) in the second term and rewriting the first term using reciprocal identities.

1/cos(x)⋅1/sin(x)+1/sin(x)

9. Apply the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

sec(x)*csc(x)+csc(x)

10. Factor out the common term csc(x) to match the right side of the identity.

csc(x)*(sec(x)+1)

Final Answer

tan(x)/(1−cos(x))=csc(x)*(1+sec(x))

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