To simplify the expression (1 + sinx) / (2 cosx + sinx), we can first multiply the numerator and denominator by the conjugate of the denominator, which is (2 cosx - sinx):
((1 + sinx) / (2 cosx + sinx)) * ((2 cosx - sinx) / (2 cosx - sinx))= (2 cosx - sinx + 2 cosx sinx - sinx^2) / (4 cos^2 x - sin^2 x)= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / (4 cos^2 x - sin^2 x)= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / (cos^2 x - sin^2 x)
Now, we can use the trigonometric identity cos^2x - sin^2x = cos2x to simplify further:
= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / cos2x= (2 cosx - sinx + sin2x) / cos2x
Therefore, the simplified expression is (2 cosx - sinx + sin2x) / cos2x.
To simplify the expression (1 + sinx) / (2 cosx + sinx), we can first multiply the numerator and denominator by the conjugate of the denominator, which is (2 cosx - sinx):
((1 + sinx) / (2 cosx + sinx)) * ((2 cosx - sinx) / (2 cosx - sinx))
= (2 cosx - sinx + 2 cosx sinx - sinx^2) / (4 cos^2 x - sin^2 x)
= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / (4 cos^2 x - sin^2 x)
= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / (cos^2 x - sin^2 x)
Now, we can use the trigonometric identity cos^2x - sin^2x = cos2x to simplify further:
= (2 cosx - sinx + 2 sinx cosx - sin^2 x) / cos2x
= (2 cosx - sinx + sin2x) / cos2x
Therefore, the simplified expression is (2 cosx - sinx + sin2x) / cos2x.