Question

1-√2 sin 45/1+√2 sin 45​

Answer 1

I am going to assume that you know that the degree measure of the angles in any triangle sum to 180 degrees. In a right triangle, one angle is, by definition, 90 degrees so the two acute angles sum to 180- 90= 90 degrees.

To prove that sin(45 (degrees)) is 1√212, consider that if a right triangle has one angle with measure 45 degrees then the other acute angle is 90- 45= 45 degrees also. From that, it follows that the two legs have the same length. Let the two legs have length s. Then by the Pythagorean theorem, the hypotenuse has length, c, satisfying c2=s2+s2=2s2c2=s2+s2=2s2. Divide both sides by 2 and c2c2 to get s2c2=12s2c2=12. Taking the positive square root of both sides gives sin(45)=sc=1√2sin(45)=sc=12.

To show that 1√2=√2212=22, multiply both numerator and denominator of 1√212 by √22:

1√2√2√2=√2(√2)2=√221222=2(2)2=22.