Factor cos(x)-sin(x)^2-1

Problem

cos(x)−sin2(x)−1

Solution

1. Apply the Pythagorean identity to rewrite sin2(x) in terms of cos(x) using sin2(x)=1−cos2(x)

cos(x)−(1−cos2(x))−1

2. Distribute the negative sign and combine like terms to simplify the expression.

cos(x)−1+cos2(x)−1

cos2(x)+cos(x)−2

3. Substitute u=cos(x) to treat the expression as a quadratic trinomial in the form u2+u−2

u2+u−2

4. Factor the quadratic by finding two numbers that multiply to −2 and add to 1 which are 2 and −1

(u+2)*(u−1)

5. Substitute back cos(x) for u to obtain the final factored form.

(cos(x)+2)*(cos(x)−1)

Final Answer

cos(x)−sin2(x)−1=(cos(x)+2)*(cos(x)−1)

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