the original problem was:
(sin x + cos x)^2 + (sin x - cos x)^2 = 2
steps too please
I got 1 for (sin x + cos x)^2
but then what does (sin x - cos x)^2 become since it's minus?
see reply to your earlier post of this question
(a+b)^2=a^2+2*a*b+b^2
(a-b)^2=a^2-2*a*b+b^2
[sin(x)+cos(x)]^2=
[sin(x)]^2 +2*sin(x)*cos(x)+ [cos(x)]^2
[sin(x)-cos(x)]^2=
[sin(x)]^2 -2*sin(x)*cos(x)+ [cos(x)]^2
[sin(x)+cos (x)]^2+[sin(x)-cos (x)]^2=
[sin(x)]^2 +2*sin(x)*cos(x)+ [cos(x)]^2+
[sin(x)]^2 -2*sin(x)*cos(x)+ [cos(x)]^2=
+[cos(x)]^2+[cos(x)]^2+[sin(x)]^2+[cos(x)]^2=
2*[[sin(x)]^2+[cos(x)]^2]=2*1=2
Becouse: [sin(x)]^2+[cos(x)]^2=1
[sin(x)+cos(x)]^2=
[sin(x)]^2 +2*sin(x)*cos(x)+ cos(x)]^2
[sin(x)-cos(x)]^2=
[sin(x)]^2 -2*sin(x)*cos(x)+ cos(x)]^2
To solve the equation (sin x + cos x)^2 + (sin x - cos x)^2 = 2, let's break it down step by step:
Step 1: Expand the squares.
First, let's expand (sin x + cos x)^2 using the formula for squaring a binomial:
(sin x + cos x)^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x)
Step 2: Simplify the expanded expression.
(sin x + cos x)^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x) becomes
sin^2(x) + 2sin(x)cos(x) + cos^2(x) + sin^2(x) - 2sin(x)cos(x) + cos^2(x)
Step 3: Combine like terms.
sin^2(x) + sin^2(x) + 2sin(x)cos(x) - 2sin(x)cos(x) + cos^2(x) + cos^2(x) simplifies to
2sin^2(x) + 2cos^2(x)
Step 4: Simplify further.
Since sin^2(x) + cos^2(x) = 1 (from the Pythagorean identity), we can substitute and simplify the equation:
2sin^2(x) + 2cos^2(x) = 2(sin^2(x) + cos^2(x)) = 2(1) = 2
Therefore, the equation becomes 2 = 2.
Step 5: Draw the conclusion.
Since the equation 2 = 2 is always true, this means that the given equation (sin x + cos x)^2 + (sin x - cos x)^2 = 2 holds for all values of x.
To answer your second question, let's handle (sin x - cos x)^2:
When squaring a binomial, you square both the first and second terms and then multiply them by two times their cross product.
Therefore, (sin x - cos x)^2 = sin^2(x) - 2sin(x)cos(x) + cos^2(x).
Keep in mind that expanding and simplifying expressions may vary and depend on specific mathematical techniques or identities used.