Question
Which choice completes the identity? \( 1-\cos ^{2} x= \) 1 \( \sin ^{2} x \) \( \sec ^{2} x \) \( \tan ^{2} x \) \( \cot ^{2} x \)
Ask by Rose Gibbs. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 1 - \cos^{2} x = \sin^{2} x \)
Solution
The Pythagorean identity states that sin²x + cos²x = 1. If you subtract cos²x from both sides of the equation, you get:
sin²x = 1 - cos²x
So, the expression 1 - cos²x is equivalent to sin²x, making that the correct choice.
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Simplify this solution Beyond the Answer
The choice that completes the identity \( 1 - \cos^{2} x = \) is \( \sin^{2} x \). This follows directly from the Pythagorean identity in trigonometry, which states that \( \sin^{2} x + \cos^{2} x = 1 \). By rearranging this identity, we see that \( 1 - \cos^{2} x \) simplifies to \( \sin^{2} x \). Now, if you think of a right triangle, the lengths of the sides correspond to sine and cosine values. Whenever you have a right triangle, knowing one angle allows you to calculate the other angle and the length of any side, giving real-world applications in fields like architecture and engineering. Just imagine plotting those angles and sides - geometry comes alive!
