Question

If $\cos \theta -\sin \theta =\frac{1}{5}$, where $0<\theta <\frac{\pi }{2}$, then the value of $5(\sin \theta +\cos \theta )$ is

Answer 1

$\cos \theta -\sin \theta =\frac{1}{5}$
Squaring both sides, we get
$1-2\sin \theta \cos \theta =\frac{1}{25}\Rightarrow 2\sin \theta \cos \theta =\frac{24}{25}$
Now,
$(\sin \theta +\cos \theta {)}^2=1+2\sin \theta \cos \theta =1+\frac{24}{25}=\frac{49}{25}\Rightarrow \sin \theta +\cos \theta =\pm \frac{7}{5}$
As $\theta \in (0,\frac{\pi }{2})$, so
$\sin \theta +\cos \theta =\frac{7}{5}\therefore 5(\sin \theta +\cos \theta )=7$


Alternate method:
$\cos \theta -\sin \theta =\frac{1}{5}$
Assuming $\sin \theta +\cos \theta =x$
Squaring and adding both the equations, we get
$1-2\sin \theta \cos \theta +1+2\sin \theta \cos \theta =x^2+\frac{1}{25}\Rightarrow x^2+\frac{1}{25}=2\Rightarrow x=\pm \frac{7}{5}$
As $\theta \in (0,\frac{\pi }{2})$, so
$\sin \theta +\cos \theta =\frac{7}{5}\therefore 5(\sin \theta +\cos \theta )=7$