Math Problem Statement
(cota-tana)/(cota+tana)=cos(2a) a = alpha prove the identity
Solution
We are tasked with proving the trigonometric identity:
Step 1: Express cotangent and tangent in terms of sine and cosine
First, recall that:
\quad \text{and} \quad \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$$ Now substitute these into the left-hand side of the equation: $$\frac{\frac{\cos(\alpha)}{\sin(\alpha)} - \frac{\sin(\alpha)}{\cos(\alpha)}}{\frac{\cos(\alpha)}{\sin(\alpha)} + \frac{\sin(\alpha)}{\cos(\alpha)}}$$ ### Step 2: Simplify the expression We simplify both the numerator and the denominator by getting a common denominator within each. #### Numerator: $$\frac{\frac{\cos^2(\alpha) - \sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}}$$ #### Denominator: $$\frac{\frac{\cos^2(\alpha) + \sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}}$$ ### Step 3: Cancel common terms Since both the numerator and denominator have the common factor $$\frac{1}{\sin(\alpha)\cos(\alpha)}$$, these factors cancel out, and we are left with: $$\frac{\cos^2(\alpha) - \sin^2(\alpha)}{\cos^2(\alpha) + \sin^2(\alpha)}$$ ### Step 4: Use the Pythagorean identity Recall the Pythagorean identity: $$\cos^2(\alpha) + \sin^2(\alpha) = 1$$ Thus, the expression simplifies to: $$\frac{\cos^2(\alpha) - \sin^2(\alpha)}{1}$$ This is simply: $$\cos(2\alpha)$$ ### Conclusion: We have shown that the left-hand side of the original equation simplifies to $$\cos(2\alpha)$$, so the identity is proven: $$\frac{\cot(\alpha) - \tan(\alpha)}{\cot(\alpha) + \tan(\alpha)} = \cos(2\alpha)$$ Would you like more details or have any questions? Here are five related questions to explore further: 1. How can you derive other double-angle identities like $$\sin(2\alpha)$$ or $$\tan(2\alpha)$$? 2. What is the geometric interpretation of trigonometric identities? 3. Can the identity be proven using a different method, such as using Euler's formula? 4. How does this identity help in solving trigonometric equations? 5. How are identities involving $$\cot$$ and $$\tan$$ useful in calculus? **Tip:** Mastering the double-angle identities like $$\cos(2\alpha)$$ helps simplify many advanced trigonometric problems in calculus and physics.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Trigonometric Identities
Double Angle Formulas
Formulas
cot(alpha) = cos(alpha) / sin(alpha)
tan(alpha) = sin(alpha) / cos(alpha)
cos^2(alpha) + sin^2(alpha) = 1
cos(2alpha) = cos^2(alpha) - sin^2(alpha)
Theorems
Pythagorean Identity
Double-Angle Identity
Suitable Grade Level
Grades 11-12
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