Questions: Verify that each of the following equations is an identity. a. (1-sin theta=fraccos ^2 theta1+sin theta)

Solution Steps

To verify that the given equation is an identity, we need to show that both sides of the equation are equal for all values of \(\theta\) for which the expressions are defined. We can start by simplifying one or both sides of the equation using trigonometric identities, such as the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).

We start with the equation we want to verify as an identity: \[ 1 - \sin \theta = \frac{\cos^2 \theta}{1 + \sin \theta} \]

We will simplify both sides of the equation. The left side is already in a simple form: \[ \text{Left Side} = 1 - \sin \theta \]

For the right side, we can use the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\): \[ \text{Right Side} = \frac{1 - \sin^2 \theta}{1 + \sin \theta} \]

The expression \(1 - \sin^2 \theta\) can be factored as \((1 - \sin \theta)(1 + \sin \theta)\): \[ \text{Right Side} = \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 + \sin \theta} \]

Assuming \(1 + \sin \theta \neq 0\), we can cancel \(1 + \sin \theta\) from the numerator and denominator: \[ \text{Right Side} = 1 - \sin \theta \]

Since both sides of the equation are equal: \[ 1 - \sin \theta = 1 - \sin \theta \] we conclude that the original equation is indeed an identity.

Final Answer