Problem 42 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin (\alpha+\beta)}{\cos \alpha \cos \beta}=\tan \alpha+\tan \beta$$

Step-by-Step Solution

Answer

Yes, the given identity, \(\frac{\sin (\alpha+\beta)}{\cos \alpha \cos \beta}=\tan \alpha+\tan \beta\), is indeed true. This was confirmed by applying the sine addition formula and several basic simplification steps, leading to express the left-hand side in the same form as the right-hand side, \(\tan \alpha + \tan \beta\).

Step 1: Utilize the Sine Addition Formula

First, apply the sine addition formula \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\). This modifies the left-hand side into \(\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta}\)

Step 2: Simplify the Expression

Separate the fraction into two parts: \(\frac{\sin \alpha \cos \beta}{\cos \alpha \cos \beta} + \frac{\cos \alpha \sin \beta}{\cos \alpha \cos \beta}\). This simplifies into \(\tan \alpha + \tan \beta\).

Step 3: Compare with the Right-Hand Side

Now the left-hand side is \(\tan \alpha + \tan \beta\), which is the same as the right-hand side. Consequently, the given trigonometric identity is verified.