1-2sina*cosa÷2=sin^2(45^@-a)?

1 Answer
Feb 15, 2018

See below for a possible answer.

Explanation:

sin^2(45-a)=sin(45-a)sin(45-a)

Using the angle subtraction formula for sine:

sin(45-a)sin(45-a)=(sin45cosa-sinacos45)(sin45cosa-sinacos45)

sin45=cos45=sqrt2/2 So:

(sqrt2/2cosa-sqrt2/2sina)(sqrt2/2cosa-sqrt2/2sina)

Factor out the sqrt2/2 in both expressions:

sqrt2/2(cosa-sina)sqrt2/2(cosa-sina)

Multiply the square roots and the trig expressions:

2/4(cos^2a-2sinacosa+sin^2a)

Simplify fraction and rearrange in parentheses:

1/2(sin^2a+cos^2a-2sinacosa)

sin^2a+cos^2a=1 therefore:

sin^2(45-a)=(1-2sinacosa)/2

QED

(Don't know how to get the degrees mark. I'll edit it if I figure it out, but all of the 45s above are 45degrees)