If \(\cos A + \sin A = \dfrac{5}{4}\), find \(\tan A\).
\(\dfrac{5 - \sqrt{7}}{5 + \sqrt{7}}\)
We are given \(\cos A + \sin A = \dfrac{5}{4}\) and must find \(\tan A\).
Square both sides: \((\cos A + \sin A)^2 = \cos^2 A + \sin^2 A + 2\sin A\cos A = \dfrac{25}{16}\).
Since \(\cos^2 A + \sin^2 A = 1\), we get \(1 + 2\sin A\cos A = \dfrac{25}{16}\), so \(2\sin A\cos A = \dfrac{9}{16}\).
Now build the difference: \((\cos A - \sin A)^2 = 1 - 2\sin A\cos A = 1 - \dfrac{9}{16} = \dfrac{7}{16}\), hence \(\cos A - \sin A = \dfrac{\sqrt{7}}{4}\).
Solve the two linear equations. Adding \(\cos A + \sin A = \dfrac{5}{4}\) and \(\cos A - \sin A = \dfrac{\sqrt{7}}{4}\) gives \(2\cos A = \dfrac{5 + \sqrt{7}}{4}\), so \(\cos A = \dfrac{5 + \sqrt{7}}{8}\).
Subtracting gives \(2\sin A = \dfrac{5 - \sqrt{7}}{4}\), so \(\sin A = \dfrac{5 - \sqrt{7}}{8}\).
Then \(\tan A = \dfrac{\sin A}{\cos A} = \dfrac{5 - \sqrt{7}}{5 + \sqrt{7}}\). The value of \(\tan A\) is \(\dfrac{5 - \sqrt{7}}{5 + \sqrt{7}}\).
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