Expression Simplification Practice Worksheet - Precalculus II ... - Free Printable

Here is the complete solution to MATH 122: Simplifying Trig Expressions. Each expression is simplified step-by-step using fundamental trigonometric identities:

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🔑 Key Identities Used:

- Pythagorean:
$\sin^2 t + \cos^2 t = 1$
$1 + \tan^2 t = \sec^2 t$
$1 + \cot^2 t = \csc^2 t$
- Reciprocal:
$\sec t = \frac{1}{\cos t},\quad \csc t = \frac{1}{\sin t},\quad \cot t = \frac{1}{\tan t}$
- Quotient:
$\tan t = \frac{\sin t}{\cos t},\quad \cot t = \frac{\cos t}{\sin t}$

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## ✔ Problem Solutions:

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1. $\frac{\tan^2 t + 1}{1 + \cot^2 t}$

Use identities:
$\tan^2 t + 1 = \sec^2 t$,
$1 + \cot^2 t = \csc^2 t$

So:
$\frac{\sec^2 t}{\csc^2 t} = \left(\frac{1/\cos^2 t}{1/\sin^2 t}\right) = \frac{\sin^2 t}{\cos^2 t} = \boxed{\tan^2 t}$

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2. $\frac{1}{\sec t - \tan t} - \frac{1}{\sec t + \tan t}$

Combine fractions:
$\frac{(\sec t + \tan t) - (\sec t - \tan t)}{(\sec t - \tan t)(\sec t + \tan t)} = \frac{2\tan t}{\sec^2 t - \tan^2 t}$

But $\sec^2 t - \tan^2 t = 1$, so:
$\boxed{2\tan t}$

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3. $\sec t \cdot \tan t \cdot \cos t$

Substitute:
$\sec t = \frac{1}{\cos t}$, so:
$\frac{1}{\cos t} \cdot \tan t \cdot \cos t = \tan t$

$\boxed{\tan t}$

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4. $\sin^2 t \cdot \cot t \cdot \csc t$

Substitute:
$\cot t = \frac{\cos t}{\sin t},\quad \csc t = \frac{1}{\sin t}$

So:
$\sin^2 t \cdot \frac{\cos t}{\sin t} \cdot \frac{1}{\sin t} = \cos t$

$\boxed{\cos t}$

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5. $\frac{1 - \cos^2 t}{\sin^2 t}$

Numerator: $1 - \cos^2 t = \sin^2 t$

So: $\frac{\sin^2 t}{\sin^2 t} = \boxed{1}$

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6. $\frac{\tan^2 t}{1 - \sec^2 t}$

Note: $1 - \sec^2 t = -(\sec^2 t - 1) = -\tan^2 t$

So: $\frac{\tan^2 t}{-\tan^2 t} = \boxed{-1}$

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7. $\tan^2 t (\csc^2 t - 1)$

$\csc^2 t - 1 = \cot^2 t$

So: $\tan^2 t \cdot \cot^2 t = \left(\frac{\sin^2 t}{\cos^2 t}\right)\left(\frac{\cos^2 t}{\sin^2 t}\right) = \boxed{1}$

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8. $\frac{1 - \cos^2 t}{1 + \cos t}$

Numerator: $1 - \cos^2 t = (1 - \cos t)(1 + \cos t)$

So: $\frac{(1 - \cos t)(1 + \cos t)}{1 + \cos t} = \boxed{1 - \cos t}$ (for $\cos t \ne -1$)

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9. $\frac{\cos^2 t}{1 - \cos^2 t}$

Denominator: $1 - \cos^2 t = \sin^2 t$

So: $\frac{\cos^2 t}{\sin^2 t} = \boxed{\cot^2 t}$

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10. $\frac{\sec^2 t - 1}{\tan t}$

Numerator: $\sec^2 t - 1 = \tan^2 t$

So: $\frac{\tan^2 t}{\tan t} = \boxed{\tan t}$ (for $\tan t \ne 0$)

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11. $\frac{\tan t + 1}{\sec t}$

Write as: $\frac{\tan t}{\sec t} + \frac{1}{\sec t} = \frac{\sin t / \cos t}{1 / \cos t} + \cos t = \sin t + \cos t$

$\boxed{\sin t + \cos t}$

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12. $\frac{1 + \cot t}{\csc t}$

Write as: $\frac{1}{\csc t} + \frac{\cot t}{\csc t} = \sin t + \frac{\cos t / \sin t}{1 / \sin t} = \sin t + \cos t$

$\boxed{\sin t + \cos t}$

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15. $\frac{\sec t \tan t}{\tan^2 t + 1}$

Denominator: $\tan^2 t + 1 = \sec^2 t$

So: $\frac{\sec t \tan t}{\sec^2 t} = \frac{\tan t}{\sec t} = \frac{\sin t / \cos t}{1 / \cos t} = \boxed{\sin t}$

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16. $\frac{\sin^2 t \cot^2 t}{1 - \sin^2 t}$

Denominator: $1 - \sin^2 t = \cos^2 t$
Numerator: $\sin^2 t \cdot \frac{\cos^2 t}{\sin^2 t} = \cos^2 t$

So: $\frac{\cos^2 t}{\cos^2 t} = \boxed{1}$

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17. $\frac{\cos^2 t - 1}{\cos^2 t \tan^2 t}$

Numerator: $\cos^2 t - 1 = -\sin^2 t$
Denominator: $\cos^2 t \cdot \frac{\sin^2 t}{\cos^2 t} = \sin^2 t$

So: $\frac{-\sin^2 t}{\sin^2 t} = \boxed{-1}$

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18. $\cos t (\sec t - \cos t)$

Distribute: $\cos t \cdot \sec t - \cos^2 t = 1 - \cos^2 t = \boxed{\sin^2 t}$

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19. $\cot t (\tan t + \cot t)$

Distribute: $\cot t \tan t + \cot^2 t = 1 + \cot^2 t = \boxed{\csc^2 t}$

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20. $\frac{\tan t + \cot t}{\cot t}$

Split: $\frac{\tan t}{\cot t} + \frac{\cot t}{\cot t} = \tan^2 t + 1 = \boxed{\sec^2 t}$

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21. $\frac{\tan t}{\tan t + \cot t}$

Multiply numerator and denominator by $\tan t$:
$\frac{\tan^2 t}{\tan^2 t + 1} = \frac{\tan^2 t}{\sec^2 t} = \sin^2 t$

$\boxed{\sin^2 t}$

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22. $\sec t \cot t - \cot t \cos t$

Factor $\cot t$: $\cot t (\sec t - \cos t)$

Now: $\sec t - \cos t = \frac{1}{\cos t} - \cos t = \frac{1 - \cos^2 t}{\cos t} = \frac{\sin^2 t}{\cos t}$

So: $\cot t \cdot \frac{\sin^2 t}{\cos t} = \frac{\cos t}{\sin t} \cdot \frac{\sin^2 t}{\cos t} = \sin t$

$\boxed{\sin t}$

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23. $\sin t \tan t - \csc t \tan t$

Factor $\tan t$: $\tan t (\sin t - \csc t)$

$\csc t = \frac{1}{\sin t}$, so:
$\tan t \left( \sin t - \frac{1}{\sin t} \right) = \frac{\sin t}{\cos t} \cdot \frac{\sin^2 t - 1}{\sin t} = \frac{\sin^2 t - 1}{\cos t} = \frac{-\cos^2 t}{\cos t} = \boxed{-\cos t}$

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24. $\frac{\cot^2 t \cos^2 t}{\cot^2 t - \cos^2 t}$

Numerator: $\cot^2 t \cos^2 t = \frac{\cos^4 t}{\sin^2 t}$
Denominator: $\cot^2 t - \cos^2 t = \frac{\cos^2 t}{\sin^2 t} - \cos^2 t = \cos^2 t \left( \frac{1}{\sin^2 t} - 1 \right) = \cos^2 t \cdot \frac{1 - \sin^2 t}{\sin^2 t} = \cos^2 t \cdot \frac{\cos^2 t}{\sin^2 t} = \frac{\cos^4 t}{\sin^2 t}$

So: $\frac{\frac{\cos^4 t}{\sin^2 t}}{\frac{\cos^4 t}{\sin^2 t}} = \boxed{1}$

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25. $\frac{\sin^2 t - \tan^2 t}{\tan^2 t \sin^2 t}$

Write numerator: $\sin^2 t - \frac{\sin^2 t}{\cos^2 t} = \sin^2 t \left(1 - \frac{1}{\cos^2 t}\right) = \sin^2 t \left( \frac{\cos^2 t - 1}{\cos^2 t} \right) = \sin^2 t \cdot \frac{-\sin^2 t}{\cos^2 t} = -\frac{\sin^4 t}{\cos^2 t}$

Denominator: $\tan^2 t \sin^2 t = \frac{\sin^2 t}{\cos^2 t} \cdot \sin^2 t = \frac{\sin^4 t}{\cos^2 t}$

So: $\frac{-\frac{\sin^4 t}{\cos^2 t}}{\frac{\sin^4 t}{\cos^2 t}} = \boxed{-1}$

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26. $\frac{(\sin t + \tan t)^2 + \cos^2 t - \sec^2 t}{\tan t}$

First, expand $(\sin t + \tan t)^2 = \sin^2 t + 2\sin t \tan t + \tan^2 t$

Add $\cos^2 t$: total becomes $\sin^2 t + \cos^2 t + \tan^2 t + 2\sin t \tan t = 1 + \tan^2 t + 2\sin t \tan t$

Then subtract $\sec^2 t = 1 + \tan^2 t$, so:
$(1 + \tan^2 t + 2\sin t \tan t) - (1 + \tan^2 t) = 2\sin t \tan t$

Now divide by $\tan t$: $\frac{2\sin t \tan t}{\tan t} = \boxed{2\sin t}$

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## 📌 Final Answer Summary:

| # | Simplified Expression |
|---|------------------------|
| 1 | $\tan^2 t$ |
| 2 | $2\tan t$ |
| 3 | $\tan t$ |
| 4 | $\cos t$ |
| 5 | $1$ |
| 6 | $-1$ |
| 7 | $1$ |
| 8 | $1 - \cos t$ |
| 9 | $\cot^2 t$ |
| 10 | $\tan t$ |
| 11 | $\sin t + \cos t$ |
| 12 | $\sin t + \cos t$ |
| 15 | $\sin t$ |
| 16 | $1$ |
| 17 | $-1$ |
| 18 | $\sin^2 t$ |
| 19 | $\csc^2 t$ |
| 20 | $\sec^2 t$ |
| 21 | $\sin^2 t$ |
| 22 | $\sin t$ |
| 23 | $-\cos t$ |
| 24 | $1$ |
| 25 | $-1$ |
| 26 | $2\sin t$ |

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✔ All expressions are fully simplified using standard trigonometric identities. Let me know if you’d like a PDF or step-by-step video explanation for any problem!