Question

Could you explain the trigonometric identities involving angles 'u' and 'x' in the following formulas:

1. sin²u = 2sinₓ cosₓ
2. tan²u = 2tanᵤ / (1 - tan²ᵤ)
3. tan²u = sin²₂ᵤ / cos²₂ᵤ
4. cos²₂ᵤ = cos²ᵤ - sin²ᵤ
5. cos²₂ᵤ = 2cos²ᵤ - 1
6. cos²₂ᵤ = 1 - 2sin²ᵤ"

Answer 1

Trigonometric identities involving angles 'u' and 'x' such as sin^2(u) and tan^2(u) relate to double angle formulas and transformations of basic trigonometric functions. These identities are essential for simplifying expressions and solving equations.

Step-by-step explanation:

The trigonometric identities you've listed involve angle relationships and transform expressions of sine, cosine, and tangent. Here are the correct versions:

• Double Angle Formulas: These identities express functions of double angles - for instance, 2u or 2x - in terms of single angles.
• Sine: sin^2(u) = 1 - cos(2u)/2. This is an application of the double-angle formula for cosine, manipulated to express sine squared.
• Tangent: tan^2(u) = 2tan(u) / (1 - tan^2(u)) and tan^2(u) = sin^2(u) / cos^2(u) are based on the basic definition of tangent as sine over cosine, as well as the tangent double-angle identity.
• Cosine: cos^2(u) = cos(2u) + sin^2(u), cos(2u) = 2cos^2(u) - 1, and cos(2u) = 1 - 2sin^2(u) are all variations of the cosine double-angle formula, which can be derived from the Pythagorean identity.

Each of these identities helps in simplifying trigonometric expressions and solving trigonometric equations. They are also useful in calculus for integrating or differentiating trigonometric functions.