Question 1170014
Let's solve this problem step-by-step.

**1. Find cos x:**

* We know sin²x + cos²x = 1.
* sin x = 2/7, so sin²x = (2/7)² = 4/49.
* cos²x = 1 - sin²x = 1 - 4/49 = 45/49.
* cos x = ±√(45/49) = ±(3√5)/7.
* Since cos x > 0, cos x = (3√5)/7.

**2. Find cos y:**

* We know sin²y + cos²y = 1.
* sin y = -2/5, so sin²y = (-2/5)² = 4/25.
* cos²y = 1 - sin²y = 1 - 4/25 = 21/25.
* cos y = ±√(21/25) = ±√21/5.
* Since cos y < 0, cos y = -√21/5.

**3. Find sin(x+y):**

* sin(x+y) = sin x cos y + cos x sin y
* sin(x+y) = (2/7)(-√21/5) + ((3√5)/7)(-2/5)
* sin(x+y) = (-2√21)/35 - (6√5)/35
* sin(x+y) = (-2√21 - 6√5)/35

**4. Find cos(x+y):**

* cos(x+y) = cos x cos y - sin x sin y
* cos(x+y) = ((3√5)/7)(-√21/5) - (2/7)(-2/5)
* cos(x+y) = (-3√(105))/35 + 4/35
* cos(x+y) = (4 - 3√105)/35

**5. Find tan(x+y):**

* tan(x+y) = sin(x+y) / cos(x+y)
* tan(x+y) = ((-2√21 - 6√5)/35) / ((4 - 3√105)/35)
* tan(x+y) = (-2√21 - 6√5) / (4 - 3√105)

**Final Answers:**

* cos x = (3√5)/7
* cos y = -√21/5
* sin(x+y) = (-2√21 - 6√5)/35
* cos(x+y) = (4 - 3√105)/35
* tan(x+y) = (-2√21 - 6√5) / (4 - 3√105)