Rút gọn biểu thức P = 1 + \(\frac{{\tan \left( {180^\circ - x} \right)}}{{\sin x.\sin \left( {90^\circ - x} \right)}}\) với x ∈ (0°; 90°).

P = 1 + \(\frac{{\tan \left( {180^\circ - x} \right)}}{{\sin x.\sin \left( {90^\circ - x} \right)}}\)

P = 1 + \(\frac{{ - \tan x}}{{\sin x.\cos x}}\)

P = 1 + \(\frac{{ - \frac{{\sin x}}{{\cos x}}}}{{\sin x.\cos x}}\)

P = 1 + \( - \frac{{\sin x}}{{\cos x}}.\frac{1}{{\sin x\cos x}}\)

P = \(1 - \frac{1}{{{{\cos }^2}x}}\)

P = \(\frac{{{{\cos }^2}x - 1}}{{{{\cos }^2}x}}\)

P = \(\frac{{{{\cos }^2}x - \left( {{{\sin }^2}x + {{\cos }^2}x} \right)}}{{{{\cos }^2}x}}\)

P = \(\frac{{ - {{\sin }^2}x}}{{{{\cos }^2}x}}\)

P = \( - {\tan ^2}x\).