DiffEQ: Determining the largest interval of a solution

[Kaylee!]

The question is to determine the solution to the following 1st order linear DE, along with the largest interval the solution is valid on:

$$cosx \frac{dy}{dx} + (sinx)y=1$$



Rewriting it shows it to be linear:
$$\frac{dy}{dx} + (tanx)y = secx$$

The intergrating factor is: $$e^{\int{tanx dx}} = e^{-ln|cosx|} = secx$$

Multiplying both sides of the DE by the integrating factor, and rewriting the LHS as a derivative of the product of the integrating factor and y:
$$\frac{d}{dx}[(secx)y]= sex^{2}x$$

(secx)y = tanx+c

y = sinx + c(cosx)

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Now how do I determine the interval?

 

[Kaylee!]

I'm multiplying both sides of the equation by secx, which has a domain of (-pi/2, pi/2), so that's the interval. Is this reasoning correct?