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Chapter 5: Problem 33

In Exercises \(29-36,\) 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 given expression simplifies to \( \tan 15^\circ \)

Step by step solution

01

Recognize the Given Pattern

Realize that the given expression \(\dfrac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ}\) fits into the pattern of an identity for the tangent of the difference of two angles, \(\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}\).
02

Identify the angles

From the pattern it is realized that angle A is \(45^\circ\) and angle B is \(30^\circ\). So, the given expression simplifies to \(\tan(A - B)\) which means \(\tan(45^\circ - 30^\circ)\)
03

Simplify the expression

Subtract the angles to simplify the expression. The given expression is then equivalent to \(\tan 15^\circ\)

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