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

Problem

tan(−x)*cos(x)=−sin(x)

Solution

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

tan(−x)*cos(x)=−tan(x)*cos(x)

2. Use the quotient identity to rewrite the tangent function in terms of sine and cosine.

−tan(x)*cos(x)=−sin(x)/cos(x)⋅cos(x)

3. Simplify the expression by canceling the common factor of cos(x) in the numerator and the denominator.

−sin(x)/cos(x)⋅cos(x)=−sin(x)

4. Conclude the verification by showing that the left side of the equation has been transformed into the right side.

−sin(x)=−sin(x)

Final Answer

tan(−x)*cos(x)=−sin(x)

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