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Chapter 6: Problem 67

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

Short Answer

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

Step by step solution

01

Apply cofunction identity

The cofunction identity for sine function is \(\sin(90掳 - x) = cos(x)\). So, using the cofunction identity, the expression \(\sin ^{2}(65掳)\) can be expressed as \(\cos ^{2}(25掳)\). Hence, the equation simplifies to: \(\sin ^{2}(25掳) + \cos ^{2} (25掳)\)
02

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: \(\sin ^{2}(25掳) + \cos ^{2} (25掳)\) simplifies further using the Pythagorean identity to become \(1\)
03

Present the final answer

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

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