Trigonometric Identities Review

To multiply the expression \( \frac{\frac{1}{\sin x}}{\cos x} \), we rewrite it as \( \frac{1}{\sin x} \cdot \frac{1}{\cos x} \). Thus, the correct choice is \( \frac{1}{\sin x} \cdot \frac{1}{\cos x} \).

To simplify \( \frac{\sin x}{\frac{1}{\cos x}} \), multiply by the reciprocal: \( \sin x \cdot \cos x \). Thus, the correct choice is \( \sin x \cdot \cos x \).

The common denominator for \( \frac{1}{\cos^2x} + \frac{1}{\sin^2x} \) is the product of the individual denominators, which is \( \cos^2x \cdot \sin^2x \). Thus, the correct answer is \( \cos^2x \cdot \sin^2x \).

The Pythagorean identity states \(\sin^2x + \cos^2x = 1\). The equation \(1 - \cos^2x = \sin^2x\) is derived from it. The other options are not valid variations of the Pythagorean identity.

To simplify \( \frac{\sin^2x+1}{\sin x} \), breaking the fraction apart is effective. This allows us to separate it into \( \frac{\sin^2x}{\sin x} + \frac{1}{\sin x} \), simplifying to \( \sin x + \csc x \).

To simplify \( \frac{\sin x}{\frac{1}{\cos x}} \), multiply by the reciprocal: \( \sin x \cdot \cos x \). Thus, the correct choice is \( \sin x \cdot \cos x \).

The identity sin²θ + cos²θ = 1 can be rearranged to sin²θ = 1 - cos²θ. Thus, the correct choice is 1 - cos²θ.

The identity sin²x + cos²x = 1 holds true for all x. Therefore, -(sin²x + cos²x) = -1. The correct answer is -1.

To factor x² - 17x - 60, we look for two numbers that multiply to -60 and add to -17. The numbers -20 and 3 fit this, leading to the factors (x - 20)(x + 3), which is the correct choice.

To factor 4h² - 12h + 9, we recognize it as a perfect square trinomial. It can be expressed as (2h - 3)², since (2h)² - 2(2h)(3) + 3² = 4h² - 12h + 9. Thus, the correct choice is (2h - 3)².