Problem 53
Question
Verify each identity. $$\frac{\sin \theta-\cos \theta}{\sin \theta}+\frac{\cos \theta-\sin \theta}{\cos \theta}=2-\sec \theta \csc \theta$$
Step-by-Step Solution
Verified Answer
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.
1Step 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 \(1-\frac{\cos \theta}{\sin \theta} + 1-\frac{\sin \theta}{\cos \theta}\).
2Step 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 \(1-\csc \theta + 1-\sec \theta\).
3Step 3. Combine the Terms
Upon combining similar terms we get: \(2-\csc \theta -\sec \theta\) as the simplified Left hand side.
4Step 4. Verify the Identity
After the simplification, we find that the LHS is equal to the RHS. Thus, proving the trigonometric identity.
Other exercises in this chapter
Problem 52
Verify each identity. \(\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha\) Hint: Write \(\cos 2 \alpha\) as \(\cos (\alpha+\alpha)\)
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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use the sum and difference formulas for cosines and sines
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Solve the equation on the interval \([0,2 \pi)\) $$(\tan x-1)(\cos x+1)=0$$
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Verify each identity. \(\tan 2 \alpha=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}\) IIint: Write \(\tan 2 \alpha\) as \(\tan (\alpha+\alpha)\)
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