Questions: Verify the identity. [1-fraccos ^2 theta1+sin theta=sin theta] To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step. [1-fraccos ^2 theta1+sin theta=1-frac1+sin theta]

Solution Steps

To verify the identity, we will start with the left-hand side (LHS) of the equation and simplify it step by step to see if it matches the right-hand side (RHS). We will use trigonometric identities and algebraic manipulation to achieve this.

1. Start with the LHS: \(1 - \frac{\cos^2 \theta}{1 + \sin \theta}\).
2. Use the Pythagorean identity: \(\cos^2 \theta = 1 - \sin^2 \theta\).
3. Substitute \(\cos^2 \theta\) in the LHS.
4. Simplify the expression to see if it matches \(\sin \theta\).

We begin with the left-hand side of the identity: \[ 1 - \frac{\cos^2 \theta}{1 + \sin \theta} \]

Using the Pythagorean identity, we know that: \[ \cos^2 \theta = 1 - \sin^2 \theta \] Substituting this into the expression gives: \[ 1 - \frac{1 - \sin^2 \theta}{1 + \sin \theta} \]

Now, we simplify the expression: \[ 1 - \frac{1 - \sin^2 \theta}{1 + \sin \theta} = 1 - \left(\frac{1}{1 + \sin \theta} - \frac{\sin^2 \theta}{1 + \sin \theta}\right) \] This simplifies to: \[ 1 - \frac{1}{1 + \sin \theta} + \frac{\sin^2 \theta}{1 + \sin \theta} \] Combining the terms leads to: \[ \frac{(1 + \sin \theta) - 1 + \sin^2 \theta}{1 + \sin \theta} = \frac{\sin^2 \theta + \sin \theta}{1 + \sin \theta} \] Factoring out \(\sin \theta\) gives: \[ \frac{\sin \theta(\sin \theta + 1)}{1 + \sin \theta} = \sin \theta \]

The simplified left-hand side is: \[ \sin \theta \] This matches the right-hand side of the original identity.

Final Answer