Directions: Verify the given problems.
1. cos² θ - sin² θ = 1 - 2sin² θ
sin² θ + cos² θ = 1 1) Use the Pythagorean Identity sin² θ + cos² θ = 1.
sin² θ + cos² θ = 1 2) Subtract sin² θ from both sides so that cos² θ equals 1 - sin² θ.
-sin² θ -sin² θ
cos² θ = 1 - sin² θ
cos² θ - sin² θ = 1 - 2sin² θ 3) Substitute cos² θ with 1 - sin² θ.
(1 - sin² θ) - sin² θ = 1 - 2sin² θ 4) Combine like terms.
1 - 2sin² θ = 1 - 2sin² θ
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1. cos² θ - sin² θ = 1 - 2sin² θ
sin² θ + cos² θ = 1 1) Use the Pythagorean Identity sin² θ + cos² θ = 1.
sin² θ + cos² θ = 1 2) Subtract sin² θ from both sides so that cos² θ equals 1 - sin² θ.
-sin² θ -sin² θ
cos² θ = 1 - sin² θ
cos² θ - sin² θ = 1 - 2sin² θ 3) Substitute cos² θ with 1 - sin² θ.
(1 - sin² θ) - sin² θ = 1 - 2sin² θ 4) Combine like terms.
1 - 2sin² θ = 1 - 2sin² θ
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2. 2 - sec² θ = 1 - tan² θ
1 + tan² θ = sec² θ 1) Use the Pythagorean Identity 1 + tan² θ = sec² θ.
2 - sec² θ = 1 - tan² θ 2) Substitute sec² θ with 1 + tan² θ.
2 - (1 + tan² θ) = 1 - tan² θ 3) Distribute the negative.
2 - 1 - tan² θ = 1 - tan² θ 4) Combine like terms.
1 - tan² θ = 1 - tan² θ
✔
1 + tan² θ = sec² θ 1) Use the Pythagorean Identity 1 + tan² θ = sec² θ.
2 - sec² θ = 1 - tan² θ 2) Substitute sec² θ with 1 + tan² θ.
2 - (1 + tan² θ) = 1 - tan² θ 3) Distribute the negative.
2 - 1 - tan² θ = 1 - tan² θ 4) Combine like terms.
1 - tan² θ = 1 - tan² θ
✔
