Question

If $\cos \theta +\sin \theta =\sqrt{2}$, find the value of $\cos \theta -\sin \theta$

Answer 1

$cos\theta +sin\theta =\sqrt{2}$

To find $cos\theta -sin\theta$

$cos\theta +sin\theta =\sqrt{2}$

squaring both sides

$(\sqrt{2}{)}^2=co{s}^2\theta +si{n}^2\theta +2sin\theta cos\theta$

$2=1+2sin\theta cos\theta [si{n}^2\theta +co{s}^2\theta =1]$

$sin\theta cos\theta =\frac{1}{2}$

$(cos\theta -sin\theta {)}^2=co{s}^2\theta +si{n}^2\theta -2sin\theta cos\theta$

$=1-2\times \frac{1}{2}$

$(cos\theta -sin\theta {)}^2=0$

$cos\theta -sin\theta =0$