Multiply and simplify.
(cos x - sin x)^2
since sin^2 + cos^2 = 1, you have
1 - 2 cosx sinx = 1-sin2x
To multiply and simplify (cos x - sin x)^2, we will expand it using the formula for square of a binomial.
Step 1: Square the first term (cos x) using the formula: (a - b)^2 = a^2 - 2ab + b^2.
So, (cos x)^2 = (cos x)^2 = cos^2(x).
Step 2: Multiply the two terms using the formula: (a - b)^2 = a^2 - 2ab + b^2.
So, -2(cos x)(sin x) = -2cos(x)sin(x).
Step 3: Square the second term (sin x) using the formula: (a - b)^2 = a^2 - 2ab + b^2.
So, - (sin x)^2 = -sin^2(x).
Now we can combine all three terms:
(cos x - sin x)^2 = (cos^2(x)) + (-2cos(x)sin(x)) + (-sin^2(x))
To simplify further, we can combine the like terms:
= cos^2(x) - 2cos(x)sin(x) - sin^2(x)
Since cos^2(x) - sin^2(x) = cos(2x), we can simplify it as:
= cos(2x) - 2cos(x)sin(x)
To multiply and simplify (cos x - sin x)^2, we can use the formula (a - b)^2 = a^2 - 2ab + b^2.
In this case, a = cos x and b = sin x. Therefore, we can substitute these values into the formula:
(cos x - sin x)^2 = (cos x)^2 - 2(cos x)(sin x) + (sin x)^2
Now, we can simplify each of these terms individually:
(cos x)^2 is the square of the cosine function, which can be written as cos^2(x).
(sin x)^2 is the square of the sine function, which can be written as sin^2(x).
2(cos x)(sin x) is the product of cosine and sine functions, which can be written as 2cos(x)sin(x).
Substituting these simplify expressions back into the original equation, we get:
(cos x - sin x)^2 = cos^2(x) - 2cos(x)sin(x) + sin^2(x)
This is the simplified form of (cos x - sin x)^2.