Given (1-tana) / (2 + Tana) = - 1 / 4 (1), find the value of Tan (a + π / 4) (2) find the value of (1 + sin2a) / (2cos ^ 2A + sin2a) | Math enthusiast | Mathematical art

Given (1-tana) / (2 + Tana) = - 1 / 4 (1), find the value of Tan (a + π / 4) (2) find the value of (1 + sin2a) / (2cos ^ 2A + sin2a)

(1－tanA)/(2＋tanA)=－1/4
2＋tanA=－4(1－tanA)
The solution is Tana = 2
Thus, Tan (a + π / 4) = [Tana + Tan (π / 4)] / [1-tan (π / 4) Tana] = (Tana + 1) / (1-tana) = - 3
That is: Tan (a + π / 4) = - 3
(1＋sin2A)/(2cos²A＋sin2A)
=[sin²A＋2sinAcosA＋cos²A]/[2cos²A＋2sinAcosA]
=[(sinA＋cosA)²]/[2cosA(cosA＋sinA)]
=(sinA＋cosA)/(2cosA)
=(1/2)(1＋tanA)
=3/2

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