Problem 56
Question
Derive the identity for \(\tan (\alpha-\beta)\) using $$ \tan (\alpha-\beta)=\tan [\alpha+(-\beta)] $$ After applying the formula for the tangent of the sum of two angles, use the fact that the tangent is an odd function.
Step-by-Step Solution
Verified Answer
The identity for \( \tan (\alpha-\beta) \) is \( \tan (\alpha-\beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\).
1Step 1: Use the formula for the tangent of the sum of two angles
Plug \(\alpha\) and \(-\beta\) into the formula for the sum of two angles, giving us: \(\tan (\alpha-\beta)\) = \(\frac{\tan(\alpha) + \tan(-\beta)}{1 - \tan(\alpha)\tan(-\beta)}\).
2Step 2: Recognize \(\tan\) as an odd function
Replace \(\tan(-\beta)\) with \(-\tan(\beta)\), using the property that the tangent function is an odd function. This gives: \(\tan (\alpha-\beta)\) = \(\frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\).
3Step 3: Final Step
The identity for \( \tan (\alpha-\beta) \) is therefore \( \tan (\alpha-\beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\).
Other exercises in this chapter
Problem 55
Use the given information to find the exact value of each of the following: a. \(\sin \frac{\alpha}{2}\) b. \(\cos \frac{\alpha}{2}\) c. \(\tan \frac{\alpha}{2}
View solution Problem 55
Verify each identity. \(\left(\tan ^{2} \theta+1\right)\left(\cos ^{2} \theta+1\right)=\tan ^{2} \theta+2\)
View solution Problem 56
Solve each equation on the interval \([0,2 \pi)\) $$ (2 \cos x-\sqrt{3})(2 \sin x-1)=0 $$
View solution Problem 56
Use the identities for \(\sin (\alpha+\beta)\) and \(\sin (\alpha-\beta)\) to solve Add the left and right sides of the identities and derive the product-to-sum
View solution