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Question
If sinx + cosx = a, then |sinx – cosx| = ______.
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Solution
Given that: sinx + cosx = a
(sinx + cosx)2 = a2
⇒ sin2x + cos2x + 2sinx cosx = a2
⇒ 1 + 2sinx cosx = a2
⇒ sinx cosx = `(a^2 - 1)/2` .......(i)
|sinx – cosx| = sin2x + cos2x – 2sinx cosx
= `1 - 2((a^2 - 1)/2)`
= 1 – (a2 – 1)
= 1 – a2 + 1
= 2 – a2
∴ |sinx – cosx| = `sqrt(2 - a^2)`
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