Final answer:
a) sin 10° = cos 40º is proven true. b) sin 30° = cos 20º is proven false. c) sin 10° = sin 30° is proven false. d) cos 20º = cos 40º is proven true.
Step-by-step explanation:
To prove each statement, we will use the given equation: sin 10° cos 20° sin 30° cos 40º = 1/16.
a) To prove sin 10° = cos 40º, we will divide both sides of the equation by sin 20° cos 30°, which equals 1/4. This gives us sin 10° = (1/16) / (1/4) = 1/4. Therefore, sin 10° equals 1/4, which is true.
b) To prove sin 30° = cos 20º, we will divide both sides of the equation by sin 10° cos 40°, which equals 1/16. This gives us sin 30° = (1/16) / (1/16) = 1. Therefore, sin 30° equals 1, which is not true.
c) To prove sin 10° = sin 30°, we will divide both sides of the equation by cos 20° cos 40°, which equals 1/4. This gives us sin 10° = (1/16) / (1/4) = 1/4. Therefore, sin 10° equals 1/4, which is not equal to sin 30°. Therefore, this statement is not true.
d) To prove cos 20° = cos 40º, we will divide both sides of the equation by sin 10° sin 30°, which equals 1/4. This gives us cos 20° = (1/16) / (1/4) = 1/4. Therefore, cos 20° equals 1/4, which is true.