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Questions and Answers
Which trigonometric identity can be used to simplify the expression $2 ext{sin}^2x + 2 ext{cos}^2x$?
Which trigonometric identity can be used to simplify the expression $2 ext{sin}^2x + 2 ext{cos}^2x$?
- $ ext{tan}^2x$
- $ ext{sin}^2x + ext{cos}^2x$
- $1$ (correct)
- $2$
Which Pythagorean identity can be used to prove the equation $ ext{tan}^2x + 1 = ext{sec}^2x$?
Which Pythagorean identity can be used to prove the equation $ ext{tan}^2x + 1 = ext{sec}^2x$?
- $ ext{sin}^2x + ext{cos}^2x = 1$ (correct)
- $ ext{csc}^2x = ext{cot}^2x + 1$
- $ ext{tan}^2x = ext{sec}^2x - 1$
- $ ext{sin}^2x = 1 - ext{cos}^2x$
Which of the following is a Pythagorean identity?
Which of the following is a Pythagorean identity?
- $rac{ ext{cos}x}{1 + ext{sin}x}$
- $ ext{sin}^2x - ext{cos}^2x$
- $rac{ ext{sin}x}{ ext{cos}x}$
- $1 - ext{cot}^2x$ (correct)
What is the simplified form of the expression $rac{ ext{cot}^2x - 1}{ ext{cot}^2x + 1}$ using Pythagorean identities?
What is the simplified form of the expression $rac{ ext{cot}^2x - 1}{ ext{cot}^2x + 1}$ using Pythagorean identities?
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Study Notes
Trigonometric Identities and Simplifications
- The expression (2\text{sin}^2x + 2\text{cos}^2x) can be simplified using the identity ( \text{sin}^2x + \text{cos}^2x = 1 ).
- By factoring out 2, the expression simplifies to (2(\text{sin}^2x + \text{cos}^2x) = 2 \cdot 1 = 2).
Proving Pythagorean Identity
- The identity ( \text{tan}^2x + 1 = \text{sec}^2x) can be proven using the fundamental Pythagorean identity ( \text{sin}^2x + \text{cos}^2x = 1 ).
- Since ( \text{tan}x = \frac{\text{sin}x}{\text{cos}x} ) and ( \text{sec}x = \frac{1}{\text{cos}x} ), the relationship holds when applying definitions of tangent and secant.
Recognizing Pythagorean Identities
- A key Pythagorean identity includes:
- ( \text{sin}^2x + \text{cos}^2x = 1 )
- Other derived forms involve transformations of this identity.
Simplification of Cotangent Expression
- The expression ( \frac{\text{cot}^2x - 1}{\text{cot}^2x + 1} ) can be simplified using Pythagorean identities.
- Recognizing that ( \text{cot}^2x = \frac{\text{cos}^2x}{\text{sin}^2x} ) leads to simplifying the expression to ( \frac{\text{cos}^2x - \text{sin}^2x}{\text{cos}^2x + \text{sin}^2x} ).
- This simplifies further to ( \frac{\text{cos}^2x - \text{sin}^2x}{1} = \text{cos}^2x - \text{sin}^2x ).
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