how does this work?
(Sin2x+cos2x+2sin²x)/(sinx+cosx)
=
(sinx+cosx
To simplify the expression (Sin2x+cos2x+2sin²x)/(sinx+cosx), we can start by rewriting it using trigonometric identities.
1) Recall the double angle identity for sine: Sin2x = 2sinx*cosx.
2) Similarly, the double angle identity for cosine is: cos2x = cos^2x - sin^2x.
Using these identities, we can rewrite the numerator of the expression as:
2sinx*cosx + cos^2x - sin^2x + 2sin²x.
Now, let's simplify further:
2sinx*cosx + cos^2x - sin^2x + 2sin²x = 2sinx*cosx + cos^2x + 2sin²x - sin^2x.
Next, let's simplify the denominator (sinx + cosx) into a common form:
sinx + cosx = (sinx + cosx) * 1 = (sinx + cosx) * (cosx/cosx) = (sinx*cosx + cos^2x) / cosx.
Now, we can substitute this back into the original expression:
(2sinx*cosx + cos^2x + 2sin²x - sin^2x) / (sinx + cosx)
= (2sinx*cosx + cos^2x + 2sin²x - sin^2x) * (cosx / (sinx*cosx + cos^2x))
= (2sinx*cosx + cos^2x + 2sin²x - sin^2x * cosx) / (sinx*cosx + cos^2x).
In this form, the expression is simplified.