Question

Sin(x+y)-sin(x-y)/cos(x+y)+cos(x-y)

Answer 1

Sinx*Cosy+Siny*Cosx-Sinx*Cosy+Siny*Cosx/
/Cosx*Cosy-Sinx*Siny+Cosx*Cosy+Sinx*Siny=
2Siny*Cosx/2Sinx*Siny=Cosx/Sinx=Ctgx

Answer 2

2sinxcosy/2cosxcosy=tgx

Answer 3

To simplify the given expression:

sin(x + y) - sin(x - y) / cos(x + y) + cos(x - y)

First, let's work with the numerator:

sin(x + y) - sin(x - y)

Using the trigonometric identity for the difference of two sines:

sin(a) - sin(b) = 2 * cos((a + b) / 2) * sin((a - b) / 2)

Here, a = x + y and b = x - y:

sin(x + y) - sin(x - y) = 2 * cos((x + y + x - y) / 2) * sin((x + y - x + y) / 2)

simplify:

sin(x + y) - sin(x - y) = 2 * cos(x) * sin(y)

Now, let's work with the denominator:

cos(x + y) + cos(x - y)

Using the trigonometric identity for the sum of two cosines:

cos(a) + cos(b) = 2 * cos((a + b) / 2) * cos((a - b) / 2)

Here, a = x + y and b = x - y:

cos(x + y) + cos(x - y) = 2 * cos((x + y + x - y) / 2) * cos((x + y - x + y) / 2)

simplify:

cos(x + y) + cos(x - y) = 2 * cos(x) * cos(y)

Now, the original expression becomes:

(2 * cos(x) * sin(y)) / (2 * cos(x) * cos(y))

The 2's in the numerator and denominator cancel out:

cos(x) * sin(y) / (cos(x) * cos(y))

Now, the cos(x) terms also cancel out:

sin(y) / cos(y)

Finally, using the trigonometric identity for the tangent function:

tan(y)

So, the simplified expression is:

tan(y)