Solve: (tanx − sin x sin y)dx + cos x cos y dy = 0

We will be using trigonometric identities and integration to solve this.

Answer: The solution of (tanx − sin x sin y)dx + cos x cos y dy = 0 is [(cos x) (sin y)] - ln(cos x) = C

Let's solve this step by step.

Explanation:

Given that, (tanx − sin x sin y)dx + cos x cos y dy = 0

Simplify it:

tanx dx − sin x sin y dx + cos x cos y dy = 0

(sin x / cos x) dx + (sin y) (- sin x dx) + (cos x) (cos y dy) = 0

We know that d(cos x) = - sin x dx, d(sin y) = cos y dy, and d(-ln cos x) = (sin x / cos x) dx

Substitue this above: we get:

d(-ln cos x) + (sin y) d(cos x) + (cos x) d(sin y) = 0

We know d[(cos x) (sin y)] = (sin y) d(cos x) + (cos x) d(sin y)

d(-ln cos x) + d[(cos x) (sin y)] = 0

Integrate on both sides:

[-ln (cos x)] + [(cos x) (sin y)] = C

[(cos x) (sin y)] - ln(cos x) = C

Hence, [(cos x) (sin y)] - ln(cos x) = C