Problem 46 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$1-\tan ^{2} 10^{\circ}+\csc ^{2} 80^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the given expression is \(2\)

Step 1: Identification

The expression given is \(1-\tan^{2} 10^{\circ}+ \csc^{2} 80^{\circ}\). Notice that \(10^{\circ}\) and \(80^{\circ}\) are complementary angles, i.e., they add up to \(90^{\circ}\). So, replacing \(80^{\circ}\) with its complementary angle \(10^{\circ}\) will make it easier to simplify the expression.

Step 2: Using the Co-function Identity

Using the Co-function identity, \(\csc(90^{\circ}-A) = \sec A\), we can convert \(\csc^{2} 80^{\circ}\) into \(\sec^{2} 10^{\circ}\). Now, our expression is \(1-\tan^{2} 10^{\circ}+ \sec^{2} 10^{\circ}\).

Step 3: Using the Pythagorean Identity

We can now use the Pythagorean identity, \(\sec^{2} A - \tan^{2} A = 1\), to simplify the expression to \(1+(1)=2\).