Problem 17 - Precalculus Essentials

Question

Verify each identity. $$\sin t \tan t=\frac{1-\cos ^{2} t}{\cos t}$$

Step-by-Step Solution

Answer

The given trigonometric identity, \(\sin t \tan t = \frac{1-\cos ^{2} t}{\cos t}\), is verified as correct by applying the substitution method and Pythagorean trigonometric identity.

Step 1: Recognize Trig Identities

By recognizing the identities, notice that \(\tan t\) can be written as the ratio of \(\sin t\) to \(\cos t\). Hence, substitute \(\tan t\) with \(\frac{\sin t}{\cos t}\) on the left side of the equation.

Step 2: Simplify the Expression

After the substitution, the left side becomes \(\sin t \cdot \frac{\sin t}{\cos t}\). By reducing the expression, it simplifies to \(\frac{\sin^{2} t}{\cos t}\).

Step 3: Apply Trig Identity

Apply the Pythagorean identity which states that \(\sin^2 t + \cos^2 t = 1\). By rearranging the formula, \(\sin^2 t\) can be expressed as \(1 - \cos^2 t\). Substitute \(\sin^2 t\) with \(1 - \cos^2 t\) in the equation.

Step 4: Verify the Identity

Once the substitution is done, the left side of the equation becomes \(\frac{1-\cos^{2} t}{\cos t}\) which exactly the same as the right side of the given identity, thus proving the identity as correct.