Question

If $\sin \theta =\cos \theta$, $\theta$ being accute, then find the value of $2\tan \theta +{\cos }^2\theta$.

Answer 1

$\sin \theta =\cos \theta$(Given)

To find: $2\tan \theta +{\cos }^2\theta$

$\Rightarrow \sin \theta =\cos \theta$

$\sin \theta =\sin ({90}^o-\theta )$ $(\because \cos \theta =\sin ({90}^o-\theta \left)\right)$

$\theta ={90}^o-\theta$

$2\theta ={90}^o$

$\Rightarrow \theta ={45}^o$

$2\tan \theta +{\cos }^2\theta =2\times \tan {45}^o+{\cos }^2{45}^o$

$=2\times 1+{\left(\frac{1}{\sqrt{2}}\right)}^2$

$=2+\frac{1}{2}=\frac{4+1}{2}=\frac{5}{2}$.