Question

PROVE THAT:cos 20° - sin 20° = ​​ \sqrt{2}sin25°​

Answer 1

Answer:

See below.

Step-by-step explanation:

$\cos(20)-\sin(20)=\sqrt{2}\sin(25)$

First, use the co-function identity:

$\sin(90-x)=\cos(x)$

We can turn the second term into cosine:

$\sin(20)=\sin(90-70)=\cos(70)$

Substitute:

$\cos(20)-\cos(70)=\sqrt{2}\sin(25)$

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

$\cos(x)-\cos(y)=-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})$

Substitute:

$\cos(20)-\cos(70)=-2\sin(\frac{20+70}{2})\sin(\frac{20-70}{2})\\\cos(20)-\cos(70) =-2\sin(45)\sin(-25)\\\cos(20)-\cos(70)=-2(\frac{\sqrt{2}}{2})\sin(-25)\\ \cos(20)-\cos(70)=-\sqrt{2}\sin(-25)$

Use the even-odd identity:

$\sin(-x)=-\sin(x)$

Therefore:

$\cos(20)-\cos(70)=-\sqrt{2}\sin(-25)\\\cos(20)-\cos(70)=-\sqrt{2}\cdot-\sin(25)\\\cos(20)-\cos(70)=\sqrt{2}\sin(25)$

Replace the second term with the original term:

$\cos(20)-\sin(20)=\sqrt{2}\sin(25)$

Proof complete.