\( \sin ^{2} \theta \) in \( \left(180^{\circ}-\theta\right) \cos \left(90^{\circ}+\theta\right)+\tan \)

Hello! It looks like you're working with some trigonometric expressions involving \( \sin^2 \theta \), \( \cos(90^{\circ} + \theta) \), and \( \tan \). However, the expression you've provided seems incomplete or a bit unclear. Let me try to interpret and assist you based on what you've shared. ### Provided Expression You wrote: \[ \sin^{2} \theta \quad \text{in} \quad \left(180^{\circ} - \theta\right) \cos \left(90^{\circ} + \theta\right) + \tan \] ### Possible Interpretation It appears you might be looking to simplify or relate the expression \( \sin^2 \theta \) within or in relation to the following trigonometric expression: \[ \left(180^{\circ} - \theta\right) \cos \left(90^{\circ} + \theta\right) + \tan \theta \] *(Assuming that \( \tan \) is \( \tan \theta \))* ### Let's Break It Down 1. **Simplify \( \cos(90^{\circ} + \theta) \):** Using the cosine addition formula: \[ \cos(90^{\circ} + \theta) = -\sin \theta \] So, the expression becomes: \[ \left(180^{\circ} - \theta\right) (-\sin \theta) + \tan \theta = -\left(180^{\circ} - \theta\right) \sin \theta + \tan \theta \] 2. **Understanding the Angle Measurement:** Notably, \( 180^{\circ} - \theta \) is an angle, but multiplying it directly by \( \sin \theta \) (which is dimensionless) is unconventional unless there's specific context, such as in a physical application where angles might be related to lengths or other quantities. 3. **Relating to \( \sin^2 \theta \):** If you're aiming to express \( \sin^2 \theta \) in terms of the given expression, we might need additional information or context. However, here's a common identity involving \( \sin^2 \theta \): \[ \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \] This identity is often useful in simplifying expressions or solving equations involving \( \sin^2 \theta \). 4. **Including \( \tan \theta \):** Recall that: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Depending on your goal, you might want to express everything in terms of sine and cosine or use other identities to combine or simplify the terms. ### Seeking Clarification To provide more accurate assistance, could you please clarify: - **What is the complete expression or equation you're working with?** - **What is the goal?** Are you trying to simplify, solve for \( \theta \), prove an identity, or something else? - **Any additional context or constraints?** For example, is this related to a specific problem in geometry, physics, or another field? Feel free to provide more details or correct any part of the expression if there's a typo. I'm here to help!