Final answer:
Using fundamental trigonometric identities, the expressions sin(2θ), cos(2θ), tan(θ), and cot(θ) are simplified using double-angle formulas and definitions of tangent and cotangent.
Step-by-step explanation:
The student's question involves using fundamental trigonometric identities to simplify several trigonometric expressions: sin(2θ), cos(2θ), tan(θ), and cot(θ). The resulting simplified forms for each expression can be derived using the double-angle formulas for sine and cosine, as well as the definitions of tangent and cotangent in terms of sine and cosine.
Simplifications:
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- Sin(2θ) simplifies to 2sin(θ)cos(θ) using the double-angle formula.
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- Cos(2θ) can be expressed in three ways: cos2(θ) - sin2(θ), 2cos2(θ) - 1, or 1 - 2sin2(θ), again referencing the double-angle formula.
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- Tan(θ) remains tan(θ) as it is already in its simplest form. However, it can also be written as sin(θ)/cos(θ).
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- Cot(θ) is the reciprocal of tan(θ) and simplifies to cos(θ)/sin(θ).
Understanding these identities is essential for solving various problems in trigonometry, such as verifying other identities or solving equations where such simplifications are required.