Due to length restrictions, we kindly invite to read carefully the explanation of the seven cases of trigonometric expressions.
How to prove trigonometric expressions
In this problem we have seven trigonometric expressions, whose equivalences have to proved by using algebraic and trigonometric properties. Now we proceed to show each procedure in detail:
Case 1
sin² x + cos² x Given
(y / r)² + (x' / r)² Definition of sine and cosine
(y² + x'²) / r² Addition of fractions with equal denominators
r² / r² Pythagorean theorem
1 Definition of division / Existence of multiplicative inverse / Result
Case 2
sin 2x Given
sin (x + x) Definition of addition
sin x · cos x + cos x · sin x Sine of the sum of two angles
2 · sin x · cos x Commutative and distributive properties / Definition of addition / Result
Case 3
cos 2x Given
cos (x + x) Definition of addition
cos x · cos x - sin x · sin x Cosine of the sum of two angles
cos² x - sin² x Definition of power / Result
Case 4
cos 2x Given
cos² x - sin² x Case 3
cos² x - (1 - cos² x) Case 1
cos² x - 1 + cos² x (- 1) · a = a / - (- a) = a
2 · cos² x - 1 Commutative and distributive properties / Definition of addition / Result
Case 5
cos 2x Given
cos² x - sin² x Case 3
(1 - sin² x) - sin² x Case 1
1 - sin² x - sin² x Associative property
1 - 2 · sin² x Commutative and distributive properties / Definition of subtraction / Result
Case 6
cos x = 2 · cos² 0.5x - 1 Case 4
1 + cos x = 2 · cos² 0.5x Compatibility with addition / Existence of additive inverse / Modulative property
(1 + cos x) / 2 = cos² 0.5x Compatibility with multiplication / Existence of multiplicative inverse / Modulative property
cos 0.5x = ± √[(1 + cos x) / 2] Definition of square root / Result
Case 7
cos x = 1 - 2 · sin² 0.5x Case 5
2 · sin² 0.5x = 1 - cos x Compatibility with addition / Existence of additive inverse / Modulative property
(1 - cos x) / 2 = sin² 0.5x Compatibility with multiplication / Existence of multiplicative inverse / Modulative property
sin 0.5x = ± √[(1 - cos x) / 2] Definition of square root / Result
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