Given that (cos(x)^2 + 4 sin(x)^2)/(1-sin(x)^2) = 7, show that tan(x)^2 = 3/2

First, we use 1 - sin(x)^2 = cos(x)^2 and get:

(LHS) (cos(x)^2 + 4 sin(x)^2)/(1-sin(x)^2)

= (cos(x)^2 + 4 sin(x)^2)/cos(x)^2

= 1 + 4 (sin(x)/cos(x))^2

= 1 + 4 tan(x)^2

Now we know that the left hand side is equal to 7.

Hence, 1 + 4 tan(x)^2 = 7  <=> tan(x)^2 = 3/2

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