Problem 26
Question
Verify each identity. $$\frac{\sin t}{\tan t}+\frac{\cos t}{\cot t}=\sin t+\cos t$$
Step-by-Step Solution
Verified Answer
Both sides of the given equation are equal when simplified, confirming the identity is valid.
1Step 1: Simplify tan and cot
Start by replacing the tan t and cot t with their equivalents in terms of sin t and cos t. This gives us: \[ \frac{\sin t}{\frac{\sin t}{\cos t}} + \frac{\cos t}{\frac{\cos t}{\sin t}} \]
2Step 2: Simplify fractions
Now simplify the fractions, and you'll end with:\[ \cos t + \sin t \]
3Step 3: Compare Left Side and Right Side
Now the left side of the equation matches the right side of the equation. Thus, we have successfully verified the identity, as \[ \frac{\sin t}{\tan t}+\frac{\cos t}{\cot t} = \sin t+\cos t \]
Other exercises in this chapter
Problem 25
Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin 25^{\circ} \cos 5^{\circ}+\cos 25^{\circ}
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Verify each identity. $$\frac{\cos 4 x-\cos 2 x}{\sin 2 x-\sin 4 x}=\tan 3 x$$
View solution Problem 26
In Exercises \(23-34\), verify each identity. $$(\sin \theta-\cos \theta)^{2}=1-\sin 2 \theta$$
View solution Problem 26
Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\cos 2 x=\frac{\sqrt{2}}{2}$$
View solution