If tanx/2=tanu/2 then prove that cosxcoshu=1

To prove that \( \cos x \cosh u = 1 \) given that \( \tan \frac{x}{2} = \tan \frac{u}{2} \), let's go through the steps:

### Given:
\[ \tan \frac{x}{2} = \tan \frac{u}{2} \]

### Step 1: Analyze the given condition
Since \( \tan \theta \) is a periodic function with period \( \pi \), the condition \( \tan \frac{x}{2} = \tan \frac{u}{2} \) implies:
\[ \frac{x}{2} = \frac{u}{2} + n\pi \]
for some integer \( n \). 

Thus,
\[ x = u + 2n\pi \]

This implies that \( x \) and \( u \) differ by a multiple of \( 2\pi \), but for the purposes of proving \( \cos x \cosh u = 1 \), we can consider the case where \( x = u \), without loss of generality, as the cosine function and hyperbolic cosine function are periodic.

### Step 2: Prove \( \cos x \cosh u = 1 \)
Assuming \( x = u \), we substitute into the expression:
\[ \cos x \cosh u = \cos u \cosh u \]

We know that:
\[ \cos x \cosh u = 1 \text{ if and only if } \cos u \cosh u = 1. \]

Let’s prove this identity for all real \( u \).

### Step 3: Use identity properties
Recall the identity for \( \cosh u \):
\[ \cosh u = \frac{e^u + e^{-u}}{2} \]

Substitute into the expression:
\[ \cos u \cdot \frac{e^u + e^{-u}}{2} = 1 \]

This equation holds true when \( \cos u = 1 \) and \( u = 0 \), but generally it might not hold for arbitrary \( u \). Thus, let's reconsider the original assumption that \( x = u \) is the most general case.

Given this, the identity holds specifically when \( u = 0 \), which is one specific scenario, or under the periodicity assumption. In general, \( \cos x \cosh u = 1 \) requires careful analysis of the behavior of the functions.

### Conclusion:
Under the special case where \( u = 0 \), it simplifies to \( \cos x = 1 \), and thus \( \cos x \cosh 0 = 1 \cdot 1 = 1 \), so the identity holds. However, in a broader context without constraints on \( u \), further conditions may be needed to guarantee the identity \( \cos x \cosh u = 1 \).

Let me know if you'd like to dive deeper into any specific part of this proof.

---

**Next Steps and Related Questions:**

1. Would you like to explore the behavior of the hyperbolic cosine function for different values of \( u \)?
2. Should we consider alternative cases where \( x \) and \( u \) are not equal but still satisfy the given conditions?
3. Are there other trigonometric or hyperbolic identities you would like to prove?
4. Would you like a visual demonstration of these functions' graphs to understand their behavior?
5. Are you interested in how this proof might change with different trigonometric identities?

**Tip:** Remember that hyperbolic functions are related to exponential functions, which can make them behave differently compared to standard trigonometric functions. Always check the domain and range of functions when dealing with identities like this.

Explore how to prove the identity \( \cos x \cosh u = 1 \) given the trigonometric equation \( \tan \frac{x}{2} = \tan \frac{u}{2} \). This detailed solution covers trigonometric identities and hyperbolic functions, suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation