If tan ? ( tanA tan(A?B) ? + sin 2 A sin 2 C? =1,A,B,C?(0, 2 ? then

Question

If $\frac{\tan (A - B)}{\tan A} + \frac{\sin^{2} C}{\sin^{2} A} = 1, A, B, C \in ((0, \frac{π}{2}))$ , then [2026]

Smart Achievers · Accepted Answer

(1)

$\frac{\tan A - \tan B}{(1 + \tan A \tan B) \tan A} + \frac{1 + \cot^{2} A}{1 + \cot^{2} C} = 1$

$Put \tan A = x, \tan B = y, \tan C = z$

$∴$ $\frac{x - y}{(1 + x y) x} + \frac{(x^{2} + 1) z^{2}}{x^{2} (z^{2} + 1)} = 1$

$∴$ $x (x - y) (z^{2} + 1) + z^{2} (1 + x^{2}) (1 + x y) = (1 + x y) x^{2} (1 + z^{2})$

$after solving we get$

$z^{2} = x y$ $∵$ $1 + x^{2} \neq 0$

$∴$ $\tan^{2} C = \tan A \cdot \tan B$

$∴$ $\tan A, \tan C, \tan B are in G.P.$

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