What is the formula of 1-cosx?

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Detailed Solution

The formulas involving "1 ± cos x" expressions are fundamental trigonometric identities that appear frequently in calculus, physics, and engineering applications. These formulas are particularly important for integration, solving trigonometric equations, and simplifying complex expressions.

Core Formula Categories

1. Half-Angle Formulas (Most Important)

Formula	Expression	Common Use
1 - cos x	`2sin²(x/2)`	Integration, half-angle calculations
1 + cos x	`2cos²(x/2)`	Integration, half-angle calculations
1 - cos 2θ	`2sin²θ`	Double angle applications
1 + cos 2θ	`2cos²θ`	Double angle applications

2. Pythagorean Identity Variations

Formula	Expression	Explanation
1 - cos²x	`sin²x`	From sin²x + cos²x = 1
1 - sin²x	`cos²x`	From sin²x + cos²x = 1
1 + cos²x	`1 + cos²x`	Cannot be simplified further
1 + sin²x	`1 + sin²x`	Cannot be simplified further

3. Power Reduction Formulas

Original Expression	Simplified Form	Application
cos²x	`(1 + cos 2x)/2`	Reducing powers for integration
sin²x	`(1 - cos 2x)/2`	Reducing powers for integration
1 - cos²x	`(1 - cos 2x)/2`	Alternative form of sin²x
1 + cos²x	`(3 + cos 2x)/2`	Power reduction

4. Advanced Variations

Formula Type	Expression	Result
√(1 - cos x)	`√2	sin(x/2)
√(1 + cos x)	`√2	cos(x/2)
(1 - cos x)/(1 + cos x)	`tan²(x/2)`	Weierstrass substitution
(1 + cos x)/(1 - cos x)	`cot²(x/2)`	Weierstrass substitution