Question

(1+cosx) (1+sinx) = 5/4 Find (1-cosx) and (1-sinx)

Answer 1

Let (1 + sinx)(1 + cosx) = K----(1), where K = 5/4
Let (1 - sinx)(1 - cosx) = X.----(2)

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)=>
1 + sinx + cosx + sinx.cosx = K

(2)=>
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/4) = 5/4 + X
8 + 2√(5X) = 5 + 4X
2√(5X) = 4X – 3
4(5X) = 16X^2 – 24X + 9
16X^2 – 24X – 20X + 9 = 0
16X^2 – 44X + 9 = 0

Using the quadratic formula,
X = (44 ± √(44^2 – 4.16.9))/(2.16) = (44 ± √(1936 – 576))/(32) = (44 ± √(1360))/32
X = 11/8 ±√(85)/8
X = 0.223, or X = 2.5274

But sinx.cosx = √(KX),
and |sinx| < 1 and |cosx| < 1 => KX < 1.

Therefore, X cannot equal 2.5274, so ignore this solution