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

Problem

sec(−x)*sin(x)=tan(x)

Solution

1. Apply the even-odd identity for the secant function, which states that sec(−x)=sec(x)

sec(−x)=sec(x)

2. Substitute the identity into the left side of the equation.

sec(x)*sin(x)

3. Use the reciprocal identity to rewrite sec(x) in terms of cos(x)

sec(x)=1/cos(x)

4. Multiply the expressions to combine them into a single fraction.

1/cos(x)⋅sin(x)=sin(x)/cos(x)

5. Apply the quotient identity which states that sin(x)/cos(x)=tan(x)

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

Final Answer

sec(−x)*sin(x)=tan(x)

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