Question: (1+sinx)(1+cosx)=5\4 ,then (1-sinx)(1-cosx)=?
Let (1 + sinx)(1 + cosx) = K, where K = 5/4
Let (1 - sinx)(1 - cosx) = X.
Multiplying the two equations together,
(1 + sinx)(1 – sinx)(1 + cosx)(1 – cosx) = KX
(1 – sin^2x)(1 – cos^2x) = KX
cos^2x.sin^2x = KX
cosx.sinx = √(KX)
Expanding the two equations,
1 + sinx + cosx + sinx.cosx = K
1 – sinx – cosx + sinx.cosx = X
Adding the two equations together
2 + 2.sinx.cosx = K + X
Substituting for cosx.sinx = √(KX)
2 + 2√(KX) = K + X
Setting K = 5/4,
2 + √(5X) = 5/4 + X
8 + 4√(5X) = 5 + 4X
4√(5X) = 4X – 3
16(5X) = 16X^2 – 24X + 9
16X^2 – 24X – 80X + 9 = 0
16X^2 – 104X + 9 = 0
Using the quadratic formula,
X = (104 ± √(104^2 – 4.16.9))/(2.16) = (104 ± √(10,816 – 576))/(32) = (104 ± √(10,240))/32
X = 13/4 ±√(10)
X = 0.087722, or X = 6.412278
But sinx.cosx = √(KX),
and |sinx| < 1 and |cosx| < 1 => KX < 1.
Therefore, X cannot equal 6.412278, so ignore this solution.
Answer: X = 0.087722