Which of the identities below follow from sin2θ+cos2θ=1? (Tick all that apply. See the course notes for reminders on sec, cosec and cot.) (A) 1+tan2θ=sec2θ (B) 1+cot2θ=cosec2θ (C)... Which of the identities below follow from sin2θ+cos2θ=1? (Tick all that apply. See the course notes for reminders on sec, cosec and cot.) (A) 1+tan2θ=sec2θ (B) 1+cot2θ=cosec2θ (C) sec2θ+cosec2θ=sec2θ.cosec2θ Read more

Steps to Solve

1. Identify the Pythagorean Identity The Pythagorean identity states that for any angle $\theta$, the following is true: $$ \sin^2 \theta + \cos^2 \theta = 1 $$

2. Rewrite the Pythagorean Identity From the Pythagorean identity, we can derive other identities by isolating one of the trigonometric functions. For example, isolating $\sin^2 \theta$ gives: $$ \sin^2 \theta = 1 - \cos^2 \theta $$

3. Express in Terms of Other Functions Next, express the derived identity in terms of cosecant, secant, or cotangent: Using the definitions of secant and cosecant, $$ \sec^2 \theta = 1 + \tan^2 \theta \quad \text{and} \quad \csc^2 \theta = 1 + \cot^2 \theta $$

4. Deriving Additional Identities From the identity $\sin^2 \theta + \cos^2 \theta = 1$, you can also express it as: $$ \frac{1}{\csc^2 \theta} + \frac{1}{\sec^2 \theta} = 1 $$ This showcases how the Pythagorean identity intersects with cosecant and secant.

5. Conclusion about Identities Thus, the derived identities involve the secant and cosecant and how they relate back to the Pythagorean identity, confirming identities like: $$ \sec^2 \theta - \tan^2 \theta = 1 $$ and $$ \csc^2 \theta - \cot^2 \theta = 1 $$