Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. $$\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1$$

Step-by-Step Solution

Answer

The statement \(\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ} = -1\) is true.

Step 1: Understanding the Given Equation

The equation provided is \(\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ} = -1\). This equation relates the trigonometric functions cotangent (\(\cot\)) and cosecant (\(\csc\)) at an angle of \(10^{\circ}\).

Step 2: Applying the Trigonometric Identity

By trigonometric identity, it is known that \(\cot^2(x) + 1 = \csc^2(x)\). If the equation is rearranged, we get \(\cot ^{2} x = \csc ^{2} x - 1\). This identity is true for all real numbers x. Let's see if this identity holds for an angle of \(10^{\circ}\).

Step 3: Comparing the Re-arranged Identity with the Given Equation

Comparing the re-arranged identity \(\cot ^{2} x = \csc ^{2} x - 1\) with the given equation \(\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ} = -1\), we see that both equations are identical. So, the statement \(\cot ^{2} 10^{\circ}-\csc ^{2} 10^{\circ}=-1\) is true.