Q. 13

If sin x + cos x = a, find the value of |sin x – cos x|

Given, sin x + cos x = a

To find the value of |sin x – cos x|

Consider square of |sin x – cos x|

|sin x – cos x|2 = |sin x|2 + |cos x|2 – 2|sin x| |cos x|

[using the formula (a + b)2= a2 + b2 +2 ab]

|sin x – cos x|2 = |sin x|2 + |cos x|2 – 2|sin x| |cos x|

= (sin2 x + cos2 x)–[(sin x + cos x)2 –sin2 x –cos2 x]

= (sin2 x + cos2 x)–[a2 – (sin2 x + cos2 x) ]

[using the formula sin2 x + cos2 x = 1]

= 1 – a2 + 1

= 2 – a2

|sin x – cos x|2 = 2 – a2

Taking square root on both sides.

Hence

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