Question

Prove that cos 20 degrees minus sin 20 degrees is equal to the square root of 2 Sin 25 degrees using the concept of compound angle.

Answer 1

To prove that cos 20 degrees - sin 20 degrees is equal to the square root of 2 Sin 25 degrees using the concept of compound angle, we can use the trigonometric identity cos(a + b) = cos a cos b - sin a sin b. By applying this identity and simplifying the equations, it is demonstrated that the two expressions are equal.

Step-by-step explanation:

To prove that cos 20 degrees - sin 20 degrees is equal to the square root of 2 sin 25 degrees using the concept of compound angle, we can use the trigonometric identity:
cos(a + b) = cos a cos b - sin a sin b

1. Let a = 20 degrees and b = -20 degrees, therefore a + b = 0 degrees
2. Using the identity, cos 0 = cos 20 degrees cos -20 degrees - sin 20 degrees sin -20 degrees
3. Since cos -20 degrees = cos 20 degrees and sin -20 degrees = -sin 20 degrees, the equation becomes cos 0 = cos² 20 degrees + sin² 20 degrees
4. By using the Pythagorean identity cos² 20 degrees + sin² 20 degrees = 1
5. Therefore, cos 0 = 1
6. Since cos 0 = 1, the equation can be written as:
   cos 20 degrees - sin 20 degrees = 1

On the other hand, to prove that the square root of 2 sin 25 degrees is equal to 1, we can simplify the expression:
sqrt(2 sin 25 degrees) = sqrt(2) * sqrt(sin 25 degrees)

1. Using the identity sin a = cos(90 degrees - a), sin 25 degrees can be written as cos 65 degrees
2. By substituting cos 65 degrees into the equation, we get:
   sqrt(2) * sqrt(cos 65 degrees)
3. By using the Pythagorean identity cos² 65 degrees + sin² 65 degrees = 1, we can simplify the equation to:
   sqrt(2) * sqrt(sin² 65 degrees)
4. Therefore, sqrt(2 sin 25 degrees) = sqrt(2)

In conclusion, cos 20 degrees - sin 20 degrees = sqrt(2 sin 25 degrees).