Questions: Multiply and simplify. (sin alpha+sec alpha)(sin alpha-sec alpha)

Solution Steps

To simplify the expression \((\sin \alpha + \sec \alpha)(\sin \alpha - \sec \alpha)\), we recognize it as a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = \sin \alpha\) and \(b = \sec \alpha\). Applying the formula, we get \(\sin^2 \alpha - \sec^2 \alpha\). We can further simplify using the identity \(\sec \alpha = \frac{1}{\cos \alpha}\).

We start with the expression \((\sin \alpha + \sec \alpha)(\sin \alpha - \sec \alpha)\). Using the difference of squares formula, we can rewrite this as: \[ \sin^2 \alpha - \sec^2 \alpha \]

Next, we substitute \(\sec \alpha\) with its definition: \[ \sec \alpha = \frac{1}{\cos \alpha} \] Thus, we have: \[ \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \] Substituting this into our expression gives: \[ \sin^2 \alpha - \frac{1}{\cos^2 \alpha} \]

To combine the terms, we need a common denominator, which is \(\cos^2 \alpha\): \[ \sin^2 \alpha - \frac{1}{\cos^2 \alpha} = \frac{\sin^2 \alpha \cos^2 \alpha - 1}{\cos^2 \alpha} \]

The numerator can be expressed as: \[ \sin^2 \alpha \cos^2 \alpha - 1 = -\left(1 - \sin^2 \alpha \cos^2 \alpha\right) \] This leads us to: \[ -\left(1 - \sin^2 \alpha \cos^2 \alpha\right) \]

Final Answer