Question
If (1 + sinx)(1 + cosx) = 3/2. Find (1 - sinx)(1 - cosx)
If (1 + sinx)(1 + cosx) = 3/2. Find (1 - sinx)(1 - cosx)
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Koala Tutor
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We can start by expanding the given equation:
(1 + sinx)(1 + cosx) = 3/2
1 + sinx + cosx + sinxcosx = 3/2
1 + sinx + cosx + sinxcosx - 3/2 = 0
2 + 2sinx + 2cosx + 2sinxcosx - 3 = 0
2sinx + 2cosx + 2sinxcosx - 1 = 0
Now, let's consider the expression we want to find:
(1 - sinx)(1 - cosx)
Expanding it, we get:
1 - sinx - cosx + sinxcosx
We can rewrite sinxcosx as 1/2*(2sinxcosx) and use the equation we derived above:
1 - sinx - cosx + 1/2*(2sinxcosx)
Now substituting 2sinx + 2cosx + 2sinxcosx - 1 = 0 from the equation we derived, we get:
1 - sinx - cosx + 1/2*(2sinxcosx)
= 1 - sinx - cosx + 1/2*(2sinx + 2cosx - 1)
= 1 - sinx - cosx + sinx + cosx - 1/2
= 1/2
Therefore, (1 - sinx)(1 - cosx) = 1/2.
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