Problem 27 - Precalculus Essentials

Question

Verify each identity. $$\tan t+\frac{\cos t}{1+\sin t}=\sec t$$

Step-by-Step Solution

Answer

The proof of the identity is verified because after several steps of simplification, the LHS equals to the RHS: \(\frac{1}{\cos t} = \frac{1}{\cos t}\)

Step 1: Write down the identity

In this exercise, the identity to verify is \(\tan t + \frac{\cos t}{1+\sin t} = \sec t\)

Step 2: Express all terms in sine/cosine form

Recall that \(\tan t = \frac{\sin t}{\cos t}\) and \(\sec t = \frac{1}{\cos t}\). Thus, the given identity can be rewritten as: \(\frac{\sin t}{\cos t} + \frac{\cos t}{1+\sin t} = \frac{1}{\cos t}\)

Step 3: Make the LHS same denominator

To simplify the expression, create a common denominator on the left-hand side (LHS). Multiply the first term by \((1+\sin t)\) and the second term by \(\cos t\), we get: \(\frac{\sin t \cdot (1+\sin t) + \cos^2 t}{\cos t \cdot (1+\sin t)} = \frac{1}{\cos t}\)

Step 4: Simplify expression

Now simplify the numerator of LHS to \(\sin t + \sin^2 t + \cos^2 t\). Remember that \(\sin^2 t + \cos^2 t = 1\). Combining like terms, we get: \(\frac{\sin t + 1}{\cos t \cdot (1+ \sin t)} = \frac{1}{\cos t}\)

Step 5: Simplify denominator

Finally, simplify the LHS further by canceling \((1 + \sin t)\) from the numerator and denominator, we get: \(\frac{1}{\cos t} = \frac{1}{\cos t}\)