Problem 10
Question
Verify each identity. $$\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1$$
Step-by-Step Solution
Verified Answer
The given trigonometric identity \(\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1\) is therefore verified.
1Step 1: Replace \(\sin ^{2} x\) using the Pythagorean identity
Start with the left side of the equation. \(\sin ^{2} x\) is equivalent to \(1 - \cos ^{2} x\), stemming from the Pythagorean identity \(\sin ^{2} x + \cos ^{2} x = 1\). So, we replace \(\sin ^{2} x\) with \(1 - \cos ^{2} x\), resulting in \(\cos ^{2} x - (1 - \cos ^{2} x)\).
2Step 2: Simplify the equation
Simplify the equation resulting from the previous step. \(\cos ^{2} x - (1 - \cos ^{2} x)\) simplifies to \(2 \cos ^{2} x - 1\), by distributing the minus sign to the \(1\) and the \(-\cos ^{2} x\). This resulting equation is now the same as the right side of the initial given equation.
Other exercises in this chapter
Problem 9
Verify each identity. $$\frac{\cos (\alpha-\beta)}{\cos \alpha \sin \beta}=\tan \alpha+\cot \beta$$
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Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Expr
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In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\
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Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x+2=\sqrt{3} \sin x, \quad x=\frac{\pi}{6}$$
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