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Chapter 5: Problem 26

Verify each identity. $$\frac{\sin t}{\tan t}+\frac{\cos t}{\cot t}=\sin t+\cos t$$

Short Answer

Expert verified
Both sides of the given equation are equal when simplified, confirming the identity is valid.

Step by step solution

01

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: \[ \frac{\sin t}{\frac{\sin t}{\cos t}} + \frac{\cos t}{\frac{\cos t}{\sin t}} \]
02

Simplify fractions

Now simplify the fractions, and you'll end with:\[ \cos t + \sin t \]
03

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 \[ \frac{\sin t}{\tan t}+\frac{\cos t}{\cot t} = \sin t+\cos t \]

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Most popular questions from this chapter

Solve each equation on the interval \([0,2 \pi)\) $$10 \cos ^{2} x+3 \sin x-9=0$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity.

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