Questions: Simplify the following expression to a single trig function with no fractions. 1-sin^2(x)/(sin(x) cos(x))

Solution Steps

To simplify the given trigonometric expression, we can use trigonometric identities. The numerator \(1 - \sin^2(x)\) can be rewritten using the Pythagorean identity as \(\cos^2(x)\). This allows us to simplify the expression by canceling out terms.

We start with the expression: \[ \frac{1 - \sin^2(x)}{\sin(x) \cos(x)} \] Using the Pythagorean identity, we know that: \[ 1 - \sin^2(x) = \cos^2(x) \] Thus, we can rewrite the expression as: \[ \frac{\cos^2(x)}{\sin(x) \cos(x)} \]

Next, we simplify the expression by canceling out one \(\cos(x)\) from the numerator and the denominator: \[ \frac{\cos^2(x)}{\sin(x) \cos(x)} = \frac{\cos(x)}{\sin(x)} = \cot(x) \]

Since \(\cot(x)\) is the reciprocal of \(\tan(x)\), we can express it as: \[ \cot(x) = \frac{1}{\tan(x)} \]

Final Answer