Question

Use the given identity to prove the related identity. Use the identity \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\) to prove the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

Short answer

Question: Prove the identity \(\cosh 2x=\cosh^2 x+\sinh^2 x\) using the identity \(\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\). Answer: To prove the identity \(\cosh 2x=\cosh^2 x+\sinh^2 x\), we start by setting \(x=y\) in the given identity \(\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\) and then simplify the expression. The result is \(\cosh 2x=\cosh^2 x+\sinh^2 x\), which is the desired identity.

Step-by-step solution

Use the given identity with \(x=y\)

We will start by setting \(x=y\) in the given identity \(\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\). This gives us:$$\cosh(2x)=\cosh x\cosh x+\sinh x\sinh x$$

Simplify the expression

Next, we rewrite the expression using familiar notation:$$\cosh 2x=\cosh^2 x+\sinh^2 x$$ This expression is exactly the identity that we wanted to prove: $$\cosh 2x=\cosh^2 x+\sinh^2 x$$

Key concepts

Hyperbolic Identities

Hyperbolic identities are similar to trigonometric identities but apply to hyperbolic functions like the hyperbolic sine (\(\sinh(x)\)) and hyperbolic cosine (\(\cosh(x)\)). These identities provide relationships between hyperbolic functions, making it easier to manipulate and understand them. One key identity is the angle addition formula for hyperbolic cosine: \[\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\] This mirrors the cosine angle addition formula in trigonometry but with hyperbolic functions. It's used to derive and prove other important identities. Understanding hyperbolic identities helps in solving complex problems involving hyperbolic functions, which often appear in calculus and differential equations.

• Remember, hyperbolic functions relate closely to exponential functions.
• They are used in real-world applications like calculating catenary curves and models in engineering.

Proof Techniques

Proof techniques are methods used to show the validity of mathematical statements. In this exercise, proof by substitution was used. We started with the given identity: \(\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y\). By setting \(x = y\), this simplifies to proving the double angle identity for hyperbolic cosine: \[\cosh(2x) = \cosh^2 x + \sinh^2 x\] This process involves substituting variables and simplifying expressions. It's similar to how you might apply known identities to new scenarios in algebra or calculus. This approach can also be combined with other methods such as induction or contradiction in more complex proofs.

• Substitution is a powerful way to apply known identities to find new results.
• Simplification often requires careful manipulation of algebraic expressions.

Hyperbolic Cosine

The hyperbolic cosine function, denoted as \(\cosh(x)\), is an essential hyperbolic function in mathematics. Defined by the formula: \[\cosh(x) = \frac{e^x + e^{-x}}{2}\] it represents the average of the exponential functions \(e^x\) and \(e^{-x}\). This function shares properties with the regular cosine function but applies to hyperbolic space. It's important in calculating distances and angles within hyperbolic geometries.

• \(\cosh(x)\) is an even function, meaning \(\cosh(-x) = \cosh(x)\).
• It comes into play in various fields such as physics and engineering, especially in modeling scenarios like the shape of a hanging cable or beam (catenary curves).

In calculus, \(\cosh(x)\) has direct applications in solving differential equations and in defining the shape of hyperbolic paraboloids.