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 You are making it look harder than it is... Just a 3 step problem. $$\frac{1}{1+sinx} + \frac{1}{1-sinx} = \frac{2}{cos^2(x)}$$ $$\frac{2}{(1+sinx)(1-sinx)} = ...$$ $$\frac{2}{1 - sin^2(x)} = \frac{2}{cos^2(x)} = 2sec^2(x)$$