Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the cofunction identities to evaluate the expression without using a calculator. $$\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}$$

Step-by-Step Solution

Answer

The result of the expression \(\sin ^{2} (25°)+\sin ^{2} (65°)\) is \(1\)

Step 1: Apply cofunction identity

The cofunction identity for sine function is \(\sin(90° - x) = cos(x)\). So, using the cofunction identity, the expression \(\sin ^{2}(65°)\) can be expressed as \(\cos ^{2}(25°)\). Hence, the equation simplifies to: \(\sin ^{2}(25°) + \cos ^{2} (25°)\)

Step 2: Using Pythagorean identity

The Pythagorean identity in trigonometry is expressed as \(\sin^{2}(x) + \cos^{2}(x) = 1\). Therefore, the updated equation from Step 1: \(\sin ^{2}(25°) + \cos ^{2} (25°)\) simplifies further using the Pythagorean identity to become \(1\)

Step 3: Present the final answer

After applying the cofunction identity in step 1 and the Pythagorean identity in step 2, the expression simplifies to \(1\)