Skills Review for Calculus of the Hyperbolic Functions

Learning Outcomes

• Verify the fundamental trigonometric identities

In the Calculus of the Hyperbolic Functions section, we will learn how to differentiate and integrate hyperbolic functions. Here we will review some trigonometric formulas and how to use them. Some of these formulas are identical to those used for hyperbolic functions.

Apply Trigonometric Formulas

A General Note: Sum and Difference Formulas for Cosine

The sum and difference formulas for cosine are:

\begin{align}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta\end{align}

\begin{align}\cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta\end{align}

A General Note: Sum and Difference Formulas for Sine

The sum and difference formulas for sine are:

$\sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$\sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta$

A General Note: Sum and Difference Formulas for Tangent

The sum and difference formulas for tangent are:

$\tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$

$\tan \left(\alpha -\beta \right)=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }$

How To: Given an identity, verify using sum and difference formulas

1. Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
2. Look for opportunities to use the sum and difference formulas.
3. Rewrite sums or differences of quotients as single quotients.
4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.

Example: Applying Trigonometric Formulas to Verify Trigonometric Identities

Verify the identity $\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)=2\sin \alpha \cos \beta$.

Example: Applying Trigonometric Formulas to Verify Trigonometric Identities

Verify the following identity.

$\frac{\sin \left(\alpha -\beta \right)}{\cos \alpha \cos \beta }=\tan \alpha -\tan \beta$

Try It

Verify the identity: $\tan \left(\pi -\theta \right)=-\tan \theta$.

A General Note: Double-Angle Formulas

The double-angle formulas are summarized as follows:

\begin{align}\sin \left(2\theta \right)&=2\sin \theta \cos \theta\\\text{ }\\ \cos \left(2\theta \right)&={\cos }^{2}\theta -{\sin }^{2}\theta \\ &=1 - 2{\sin }^{2}\theta \\ &=2{\cos }^{2}\theta -1 \\\text{ }\\ \tan \left(2\theta \right)&=\frac{2\tan \theta }{1-{\tan }^{2}\theta }\end{align}

Establishing identities using the double-angle formulas is performed using the same steps we used to establish identities using the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

Example: Applying Trigonometric Formulas to Verify Trigonometric Identities

Establish the following identity using double-angle formulas:

$1+\sin \left(2\theta \right)={\left(\sin \theta +\cos \theta \right)}^{2}$

Try It

Establish the identity: ${\cos }^{4}\theta -{\sin }^{4}\theta =\cos \left(2\theta \right)$.

Example: Applying Trigonometric Formulas to Verify Trigonometric Identities

Verify the identity:

$\tan \left(2\theta \right)=\frac{2}{\cot \theta -\tan \theta }$

Try It

Verify the identity: $\cos \left(2\theta \right)\cos \theta ={\cos }^{3}\theta -\cos \theta {\sin }^{2}\theta$.