The given expression is:

$(\sin \theta + 1)(\tan \theta - \sec \theta)$

Let’s simplify it step by step.

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Step 1: Expand the product

Expand the two terms using distributive property:

$(\sin \theta + 1)(\tan \theta - \sec \theta) = (\sin \theta)(\tan \theta) - (\sin \theta)(\sec \theta) + (1)(\tan \theta) - (1)(\sec \theta)$

Simplify each term:

$= \sin \theta \cdot \tan \theta - \sin \theta \cdot \sec \theta + \tan \theta - \sec \theta$

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Step 2: Substitute trigonometric identities

• Recall that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$.

Substitute these into the expression:

$= \sin \theta \cdot \frac{\sin \theta}{\cos \theta} - \sin \theta \cdot \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta}$

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Step 3: Simplify each term

• The first term becomes $\frac{\sin^2 \theta}{\cos \theta}$.
• The second term becomes $-\frac{\sin \theta}{\cos \theta}$.
• The third term remains $\frac{\sin \theta}{\cos \theta}$.
• The fourth term remains $-\frac{1}{\cos \theta}$.

Combine all terms:

$= \frac{\sin^2 \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta}$

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Step 4: Combine like terms

The $-\frac{\sin \theta}{\cos \theta}$ and $+\frac{\sin \theta}{\cos \theta}$ cancel each other out:

$= \frac{\sin^2 \theta}{\cos \theta} - \frac{1}{\cos \theta}$

Factor out $\frac{1}{\cos \theta}$:

$= \frac{1}{\cos \theta} (\sin^2 \theta - 1)$

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Step 5: Simplify using Pythagorean identity

Recall that $\sin^2 \theta - 1 = -\cos^2 \theta$. Substitute this:

$= \frac{1}{\cos \theta} (-\cos^2 \theta)$

Simplify:

$= -\cos \theta$

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Final Answer:

$(\sin \theta + 1)(\tan \theta - \sec \theta) = -\cos \theta$

Let me know if you want more details or have questions!

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Here are 5 related questions you can explore:

1. What are the key trigonometric identities used to simplify expressions?
2. How does the distributive property help in expanding products?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How can $\tan \theta$ and $\sec \theta$ be expressed in terms of $\sin \theta$ and $\cos \theta$?
5. How can this simplification be visualized on a unit circle?

Tip: Always simplify expressions step-by-step, applying one identity at a time to avoid confusion!