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

Problem

cos(x)−sec(x)=−sin(x)*tan(x)

Solution

1. Express the secant function in terms of cosine using the reciprocal identity sec(x)=1/cos(x)

cos(x)−1/cos(x)

2. Find a common denominator to combine the terms into a single fraction.

(cos2(x)−1)/cos(x)

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies cos2(x)−1=−sin2(x)

(−sin2(x))/cos(x)

4. Rewrite the expression by splitting the squared sine term to isolate the tangent function.

−sin(x)⋅sin(x)/cos(x)

5. Substitute the quotient identity sin(x)/cos(x)=tan(x) to reach the final form.

−sin(x)*tan(x)

Final Answer

cos(x)−sec(x)=−sin(x)*tan(x)

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