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For part d):

$\frac{\sin(x + y)}{\sin(x - y)} = \frac{\tan(x) + \tan(y)}{\tan(x) - \tan(y)}$

For part e):

$\frac{\sec(-t)}{\tan(-t)} = - \csc(t)$

Would you like me to help explain or simplify these identities further? Let me know how you'd like to proceed!

Here are some related questions:

1. How do you prove trigonometric identities?
2. Can you derive the formulas for $\tan(x+y)$ and $\tan(x-y)$?
3. What are the basic identities for secant and tangent functions?
4. How do you apply negative angles in trigonometric identities?
5. Can you simplify complex trigonometric expressions step by step?

Tip:

When working with trigonometric identities, always remember to check if you can simplify the expression using known formulas for addition and subtraction (like the sum or difference formulas for sine and tangent).