Important Trignometric Identities

🔹 1. Definitions of Trigonometric Functions

All trigonometric functions can be written using $\sin x$ and $\cos x$:

$$\tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}$$ $$\sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}$$

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🔹 2. Pythagorean Identities

These are the most important identities to memorize:

$$\sin^2 x + \cos^2 x = 1$$ $$1 + \tan^2 x = \sec^2 x$$ $$1 + \cot^2 x = \csc^2 x$$

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🔹 3. Power-Reducing Identities

Useful when dealing with integrals or simplifying expressions:

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

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🔹 4. When Are Functions Undefined?

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🔹 5. Key Strategy (Very Important)

💡 Most trig problems become easier if you:

• Rewrite everything in terms of $\sin x$ and $\cos x$
• Use identities to simplify before solving
• Look for patterns like $1 + \tan^2 x$ or $\sin^2 x + \cos^2 x$

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🔹 6. Example

Simplify:

$$1 + \tan^2 x$$

Solution:

Using identity:

$$1 + \tan^2 x = \sec^2 x$$

✅ Final Answer: $\sec^2 x$

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🔹 7. Example (Rewrite in sin & cos)

Simplify:

$$\frac{\tan x}{\sec x}$$

Solution:

Rewrite:

$$\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} = \sin x$$

✅ Final Answer: $\sin x$

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Quick Summary

• Everything can be rewritten using $\sin x$ and $\cos x$
• Memorize the 3 Pythagorean identities
• Use power-reducing identities for integrals
• Always simplify before solving

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