sin a + cos a = tan a
sin a + cos a = sin a / cos a
sin a .cos a + cos a .cos a = sin a
sin a .cos a + 1 - sin a .sin a = sin a
cos a = (sin a .sin a + sin a - 1)/sin a
cos a = sin a + 1 - 1/sin a
0°< a < 180° => 0 < sin a =< 1
若 a = 90°,sin a = 1,
那么 cos a = 1 + 1 - 1 = 1
矛盾!
sin a + 1 = 1/sin a
sin a .sin a + sin a = 1
sin a .sin a + sin a - 1 = 0
sin a = ( -1 + √5)/2
a = 38.17°或 141.83°
若 0°< a < 38.17° 或 141.83°< a < 180°,
sin a + 1 < 1/sin a,
cos a < 0
∵ 当 90°< a < 180°时,-1 < cos a < 0
∴ 141.83°< a < 180°
若 38.17°< a < 141.83°,
sin a + 1 > 1/sin a,
cos a > 0
∵ 当 0°< a < 90°时,0 < cos a < 1
∴ 38.17°< a < 90°
=> a 的范围：38.17°< a < 90°及 141.83°< a < 180°