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Simplify 1+tan^2x/1+cot^2x
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Interpreting the expression as (1 + tan^2 x) / (1 + cot^2 x):
1 + tan^2 x = sec^2 x and 1 + cot^2 x = csc^2 x, so
(1 + tan^2 x)/(1 + cot^2 x) = sec^2 x / csc^2 x = (1/cos^2 x)/(1/sin^2 x) = sin^2 x / cos^2 x = tan^2 x.
Note: this equality holds where the original expression is defined, i.e. sin x ≠ 0 and cos x ≠ 0 (so x is not an integer multiple of π/2).
1 + tan^2 x = sec^2 x and 1 + cot^2 x = csc^2 x, so
(1 + tan^2 x)/(1 + cot^2 x) = sec^2 x / csc^2 x = (1/cos^2 x)/(1/sin^2 x) = sin^2 x / cos^2 x = tan^2 x.
Note: this equality holds where the original expression is defined, i.e. sin x ≠ 0 and cos x ≠ 0 (so x is not an integer multiple of π/2).
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