Question

If $\sin \theta +\cos \theta =a$ then find the value of $\sin \theta -\cos \theta$.

Answer 1

Given,

$\sin \theta +\cos \theta =a$

Squaring both sides we get,

${\sin }^2\theta +{\cos }^2\theta +2\sin \theta .\cos \theta =a^2$

or, $2\sin \theta .\cos \theta =a^2-1$......(1).

We have

$(\sin \theta +\cos \theta {)}^2-(\sin \theta -\cos \theta {)}^2=4\sin \theta .\cos \theta$ [ Since $(a+b{)}^2-(a-b{)}^2=4ab$ (standard result)]

or, $a^2-(\sin \theta -\cos \theta {)}^2=2(-1+a^2)$ [ Using (1)]

or, $(\sin \theta -\cos \theta {)}^2=2-a^2$

or, $(\sin \theta -\cos \theta )=\pm \sqrt{2-a^2}$.