Been trying this forever so it says: Simplify 1+sin2x+cos2x/1+sin2x-cos2x
ive tried this problem so many times I need help with it...
I believe you mean
(1+sin2x + cos2x)/(1+sin2x - cos2x)
= (sin^2 x + cos^2 x + 2sinxcosx + cos^2 x - sin^2x)/(sin^2 x + cos^2 x + 2sinxcosx - cos^2x + sin^2x)
= (2cos^2 x + 2sinxcosx)/(2sin^2 x + 2sinxcosx)
= 2cosx(cosx+sinx)/((2sinx(sinx+cos))
= 2cosx/2sinx
= cot x
To simplify the expression (1 + sin(2x) + cos(2x)) / (1 + sin(2x) - cos(2x)), we can use the formula for sin(2x) and cos(2x).
Recall that sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x).
Let's substitute these values into the expression:
(1 + sin(2x) + cos(2x)) / (1 + sin(2x) - cos(2x)) = (1 + 2sin(x)cos(x) + cos^2(x) - sin^2(x)) / (1 + 2sin(x)cos(x) - cos^2(x) + sin^2(x))
Now, let's simplify the numerator and denominator separately:
Numerator: (1 + 2sin(x)cos(x) + cos^2(x) - sin^2(x))
= (1 + cos^2(x) - sin^2(x) + 2sin(x)cos(x))
Notice that 1 = cos^2(x) + sin^2(x), so we can substitute this value in:
= (cos^2(x) + sin^2(x) + cos^2(x) - sin^2(x) + 2sin(x)cos(x))
= (2cos^2(x) + 2sin(x)cos(x))
= 2(cos^2(x) + sin(x)cos(x))
Denominator: (1 + 2sin(x)cos(x) - cos^2(x) + sin^2(x))
= (cos^2(x) + sin^2(x) - cos^2(x) + sin^2(x) + 2sin(x)cos(x))
= 2(sin^2(x) + cos(x)sin(x))
= 2sin(x)(sin(x) + cos(x))
Now, we can simplify the expression by canceling out common factors:
(2(cos^2(x) + sin(x)cos(x))) / (2sin(x)(sin(x) + cos(x)))
= (cos^2(x) + sin(x)cos(x)) / (sin(x)(sin(x) + cos(x)))
Therefore, the expression (1 + sin(2x) + cos(2x)) / (1 + sin(2x) - cos(2x)) simplifies to (cos^2(x) + sin(x)cos(x)) / (sin(x)(sin(x) + cos(x))).
To simplify the expression 1 + sin^2x + cos^2x / 1 + sin^2x - cos^2x, we can use a trigonometric identity. In this case, we'll use the Pythagorean identity: sin^2x + cos^2x = 1.
Let's break it down step by step:
Step 1: Rewrite the expression using the Pythagorean identity.
1 + sin^2x + cos^2x / 1 + sin^2x - cos^2x
= 1 + 1 / 1 + sin^2x - cos^2x
Step 2: Simplify the denominator using the Pythagorean identity.
1 + 1 / 1 + sin^2x - cos^2x
= 1 + 1 / 1 + 1 - cos^2x - cos^2x
= 1 + 1 / 2 - cos^2x
Step 3: Simplify the numerator and denominator individually.
1 + 1 / 2 - cos^2x
= 2 / 2 - cos^2x
= 1 - cos^2x
So the simplified expression is 1 - cos^2x.