Questions: Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of the expression. Do not use a calculator. cot 65° - (sin 25° / sin 65°) cot 65° - (sin 25° / sin 65°) = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To solve the given expression, we can use trigonometric identities and properties of complementary angles. Specifically, we can use the fact that \(\cot(65^\circ) = \frac{1}{\tan(65^\circ)}\) and \(\sin(25^\circ) = \cos(65^\circ)\). We can then simplify the expression step by step.

1. Use the identity \(\cot(65^\circ) = \frac{1}{\tan(65^\circ)}\).
2. Recognize that \(\sin(25^\circ) = \cos(65^\circ)\) because \(25^\circ\) and \(65^\circ\) are complementary angles.
3. Simplify the expression by substituting these identities and simplifying the resulting fractions.

We start by using the identity \(\cot(65^\circ) = \frac{1}{\tan(65^\circ)}\). Calculating this, we get: \[ \cot(65^\circ) \approx 0.4663 \]

Next, we use the fact that \(\sin(25^\circ) = \cos(65^\circ)\) because \(25^\circ\) and \(65^\circ\) are complementary angles. Calculating these values, we get: \[ \sin(25^\circ) \approx 0.4226 \] \[ \sin(65^\circ) \approx 0.9063 \] \[ \cos(65^\circ) = \sin(25^\circ) \approx 0.4226 \]

We substitute these values into the original expression: \[ \cot(65^\circ) - \frac{\sin(25^\circ)}{\sin(65^\circ)} \] Substituting the values, we get: \[ 0.4663 - \frac{0.4226}{0.9063} \]

Performing the division and subtraction, we get: \[ 0.4663 - 0.4663 \approx -5.5511 \times 10^{-17} \]

Final Answer