Prove that cosh^2(θ) - sinh^2(θ) = 1.
Understand the Problem
The question asks to prove the hyperbolic trigonometric identity: cosh^2(θ) - sinh^2(θ) = 1. This involves using the definitions of cosh(θ) and sinh(θ) in terms of exponentials and simplifying the expression to arrive at the result of 1.
Answer
$cosh^2 \theta - sinh^2 \theta = 1$
Answer for screen readers
$cosh^2 \theta - sinh^2 \theta = 1$
Steps to Solve
- Write the definitions of $cosh(\theta)$ and $sinh(\theta)$
The hyperbolic cosine and hyperbolic sine are defined as follows: $cosh(\theta) = \frac{e^\theta + e^{-\theta}}{2}$ and $sinh(\theta) = \frac{e^\theta - e^{-\theta}}{2}$
- Substitute the definitions into the equation
Substitute these definitions into the left-hand side of the equation $cosh^2(\theta) - sinh^2(\theta)$:
$$cosh^2(\theta) - sinh^2(\theta) = \left(\frac{e^\theta + e^{-\theta}}{2}\right)^2 - \left(\frac{e^\theta - e^{-\theta}}{2}\right)^2$$
- Expand the squares
Expand each of the squared terms:
$$= \frac{(e^\theta + e^{-\theta})^2}{4} - \frac{(e^\theta - e^{-\theta})^2}{4}$$ $$= \frac{e^{2\theta} + 2e^\theta e^{-\theta} + e^{-2\theta}}{4} - \frac{e^{2\theta} - 2e^\theta e^{-\theta} + e^{-2\theta}}{4}$$
- Simplify the exponentials
Simplify the $e^\theta e^{-\theta}$ terms, knowing that $e^\theta e^{-\theta} = e^{\theta - \theta} = e^0 = 1$. $$= \frac{e^{2\theta} + 2 + e^{-2\theta}}{4} - \frac{e^{2\theta} - 2 + e^{-2\theta}}{4}$$
- Combine the fractions
Combine the two fractions into one: $$= \frac{(e^{2\theta} + 2 + e^{-2\theta}) - (e^{2\theta} - 2 + e^{-2\theta})}{4}$$
- Distribute the negative sign and simplify
Distribute the negative sign in the second term and simplify: $$= \frac{e^{2\theta} + 2 + e^{-2\theta} - e^{2\theta} + 2 - e^{-2\theta}}{4}$$ $$= \frac{4}{4}$$
- Final simplification
Simplify the fraction: $$= 1$$
$cosh^2 \theta - sinh^2 \theta = 1$
More Information
This identity is analogous to the trigonometric identity $cos^2 \theta + sin^2 \theta = 1$. Both are fundamental identities in their respective fields (hyperbolic trigonometry and standard trigonometry).
Tips
A common mistake is incorrect expansion of the squared terms, particularly forgetting the cross term (the $2ab$ term in $(a+b)^2$ or $(a-b)^2$). Another common mistake is with the exponential arithmetic; specifically, students sometimes incorrectly simplify $e^\theta e^{-\theta}$.
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