Problem 75 - Precalculus Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\tan ^{2} 15^{\circ}-\sec ^{2} 15^{\circ}=-1$$

Step-by-Step Solution

Answer

The given statement is false. The correct equation should be \(\tan^{2} 15^{\circ} - \sec^{2} 15^{\circ} = -1\)

Step 1: Testing the given equation

Begin by plugging the value of \(15^{\circ}\) into \(\tan ^{2} 15^{\circ}\) and \(\sec ^{2} 15^{\circ}\) and calculate the difference to see if it's equal to \(-1\).

Step 2: Identifying the problem

If the difference calculated in Step 1 is not equal to \(-1\), then the given equation is false. We need to identify what should be the correct equation.

Step 3: Correct the equation

To correct the equation, one must recall the Pythagorean trigonometric identity which is \(\tan^{2}\theta + 1 = \sec^{2}\theta\). Scientists can rearrange it as \(\tan^{2}\theta - \sec^{2}\theta = -1\), which is the correct equation.