$$ \sin ^{2} heta $$ in $$ \left(180^{\circ}- heta ight) \cos \left(90^{\circ}+ heta ight)+ an $$

Question

$$ \sin ^{2} 	heta $$ in $$ \left(180^{\circ}-	hetaight) \cos \left(90^{\circ}+	hetaight)+	an $$

UpStudy AI · Accepted Answer

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!
To simplify the expression $$ \sin^{2} \theta $$ within $$ \left(180^{\circ} - \theta\right) \cos \left(90^{\circ} + \theta\right) + \tan \theta $$, follow these steps: 1. **Simplify $$ \cos(90^{\circ} + \theta) $$:** $$ \cos(90^{\circ} + \theta) = -\sin \theta $$ 2. **Substitute back into the original expression:** $$ \left(180^{\circ} - \theta\right) (-\sin \theta) + \tan \theta = -\left(180^{\circ} - \theta\right) \sin \theta + \tan \theta $$ 3. **Express $$ \tan \theta $$ in terms of sine and cosine:** $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$ 4. **Combine like terms if possible:** $$ -\left(180^{\circ} - \theta\right) \sin \theta + \frac{\sin \theta}{\cos \theta} $$ 5. **Factor out $$ \sin \theta $$ if needed:** $$ \sin \theta \left[ -\left(180^{\circ} - \theta\right) + \frac{1}{\cos \theta} \right] $$ 6. **Use the identity $$ \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} $$ if necessary for further simplification.** **Final Simplified Expression:** $$ \sin \theta \left[ -\left(180^{\circ} - \theta\right) + \frac{1}{\cos \theta} \right] $$ This expression relates $$ \sin^{2} \theta $$ within the given trigonometric context.

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