PROVE THAT:

cos 20° - sin 20° = \sqrt{2}sin25°

See below.

Step-by-step explanation:

First, use the co-function identity:

We can turn the second term into cosine:

Substitute:

Now, use the sum to product formulas. We will use the following:

Substitute:

Use the even-odd identity:

Therefore:

Replace the second term with the original term:

Proof complete.

hope this !

step-by-step explanation:

d = distance between moon and sun, which in this problem is the hypotenuse of a right triangle.  

the distance y is the side opposite to angle x in the right triangle.   use the definition of the sine function from basic trigonometry.

sin(x) = y/d ⇒

d = y/sin(x)