Question

If sinx+cosx = a, then |sinx–cosx|=?​

Answer 1

Step-by-step explanation:

We have,

$\sin(x) + \cos(x) = a$

$\implies \{\sin(x) + \cos(x) \} ^{2} = a^{2}$

$\implies\sin^{2} (x) + \cos ^{2} (x) + 2 \sin(x) \cos(x) = a^{2}$

$\implies1 + 2 \sin(x) \cos(x) = a^{2}$

$\implies 2 \sin(x) \cos(x) = a^{2} - 1$

Now,

$| \sin(x) - \cos(x) | = \sqrt{ \{ \sin(x) - \cos(x) \} ^{2} }$

$\implies | \sin(x) - \cos(x) | = \sqrt{ \sin^{2} (x) + \cos^{2} (x) - 2 \sin(x) \cos(x) }$

$\implies | \sin(x) - \cos(x) | = \sqrt{1 -( {a}^{2} - 1)}$

$\implies | \sin(x) - \cos(x) | = \sqrt{1 -{a}^{2} + 1)}$

$\implies | \sin(x) - \cos(x) | = \sqrt{2 -{a}^{2} }$