Verify the Identity (tan(x))/(sec(x))=sin(x)

Problem

tan(x)/sec(x)=sin(x)

Solution

1. Express in terms of sine and cosine by using the quotient identity tan(x)=sin(x)/cos(x) and the reciprocal identity sec(x)=1/cos(x)

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

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

sin(x)/cos(x)⋅cos(x)/1

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

sin(x)

4. Verify the equality by observing that the simplified left side matches the right side of the identity.

sin(x)=sin(x)

Final Answer

tan(x)/sec(x)=sin(x)

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