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Chapter 5: Problem 33

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}$$

Short Answer

Expert verified
The simplified expression is \(\tan 15^{\circ}\).

Step by step solution

01

Recognizing the Identity

Observing the given problem, it can be noticed that the expression is in the form of a well-known trigonometric identity, \(\frac{\tan A-\tan B}{1+\tan A \tan B} = \tan (A-B)\). This identity is valid for any angles A and B, where \(\tan A\) and \(\tan B\) are defined.
02

Applying the Identity

Using the identity from Step 1, rewrite the given expression as \(\tan (45^{\circ}-30^{\circ})\).
03

Simplifying the Expression

Now simplify the expression by subtracting the angles inside the tangent function, leading to \(\tan 15^{\circ}\).

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