Question

If $\cos \theta -\sin \theta =\frac{1}{5}$where $0<\theta <\frac{\pi }{2}$

Answer 1

Given that $\cos \theta -\sin \theta =\frac{1}{5}$
On squaring, we get $1-2\sin 2\theta =\frac{1}{25}$
$\therefore \sin 2\theta =\frac{24}{25}$
Now, $\cos 2\theta =\sqrt{1-{\sin }^{2}2\theta }=\frac{7}{25}$
$\Rightarrow 2{\cos }^2\theta -1=\frac{7}{25}$
$\therefore \cos \theta =\frac{4}{5}$
$\Rightarrow \sin \theta =\frac{3}{5}$
Therefore, $\frac{\cos \theta +\sin \theta }{2}=\frac{7}{10}$