Given equation: cosylog(secx tanx)dx = cosxlog(secy tany)dy

To solve this equation, we will separate the variables and integrate both sides with respect to their respective variables.

Separating variables:
Dividing both sides of the equation by cosx log(secy tany), we get:

(cosecy dy) / log(secy tany) = (secxdx) / log(secx tanx)

Integrating both sides:
Now, we integrate both sides of the equation with respect to their respective variables:

∫(cosecy dy) / log(secy tany) = ∫(secxdx) / log(secx tanx)

Integrating the left side:
To integrate the left side, we can use the substitution method. Let's substitute u = secy tany. Then, du = (secy tany) dy.

Using this substitution, the left side of the equation becomes:

∫(cosecy dy) / log(u) = ∫(du / log(u))

Using the property of logarithms, we simplify the integral:

∫(du / log(u)) = log(log(u))

Therefore, the left side of the equation becomes log(log(u)).

Integrating the right side:
To integrate the right side, we can use the substitution method. Let's substitute v = secx tanx. Then, dv = (secx tanx) dx.

Using this substitution, the right side of the equation becomes:

∫(secxdx) / log(secx tanx) = ∫(dv / log(v))

Using the property of logarithms, we simplify the integral:

∫(dv / log(v)) = log(log(v))

Therefore, the right side of the equation becomes log(log(v)).

Re-writing the equation:
After integrating both sides, the equation becomes:

log(log(u)) = log(log(v))

Simplifying the equation:
Since both sides are equal, we can drop the logarithms:

log(u) = log(v)

This implies:

u = v

Substituting back the values of u and v:

secy tany = secx tanx

Taking the reciprocal of both sides:

cosy coty = cosx cotx

Using the identity cotx = 1/tanx:

cosy/tany = cosx/tanx

Simplifying further:

cosy/siny = cosx/cosx

This implies:

cosy = siny

Taking the reciprocal of both sides:

secy = cscy

Since secy = 1/cosy and cscy = 1/siny, this implies:

1/cosy = 1/siny

Cross-multiplying:

siny = cosy

This is only true when y = x + 2πn or y = -x + 2πn, where n is an integer.

Conclusion:
Therefore, the solution of the given equation is y = x + 2πn or y = -x + 2πn, where n is an integer. None of the given options (a), (b), or (c) match this solution, so the correct answer is option (d) "none of these