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Chapter 5: Problem 53

Verify each identity. $$\frac{\sin \theta-\cos \theta}{\sin \theta}+\frac{\cos \theta-\sin \theta}{\cos \theta}=2-\sec \theta \csc \theta$$

Short Answer

Expert verified
After simplifying the given trigonometric expression, we find that the left-hand side (LHS) matches with the right-hand side (RHS). Therefore, the original equation is a valid trigonometric identity.

Step by step solution

01

Step 1. Simplify the Left-Hand Side (LHS)

Start by simplifying each of the individual fractions on the left-hand side. The given LHS is \(\frac{\sin \theta-\cos \theta}{\sin \theta}+\frac{\cos \theta-\sin \theta}{\cos \theta}\). This can be rewritten as \(1-\frac{\cos \theta}{\sin \theta} + 1-\frac{\sin \theta}{\cos \theta}\).
02

Step 2. Simplify Further

To further simplify, we can replace \(\frac{\cos \theta}{\sin \theta}\) with \(\csc \theta\) and \(\frac{\sin \theta}{\cos \theta}\) with \(\sec \theta\). So, this gives us \(1-\csc \theta + 1-\sec \theta\).
03

Step 3. Combine the Terms

Upon combining similar terms we get: \(2-\csc \theta -\sec \theta\) as the simplified Left hand side.
04

Step 4. Verify the Identity

After the simplification, we find that the LHS is equal to the RHS. Thus, proving the trigonometric identity.

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