Question

If $\theta$ is an acute angle and $\sin \theta =\cos \theta$. The value of $2{\tan }^2\theta +{\sin }^2\theta -1$ is,

Answer 1

Find the value.

As we know that, Trigonometric functions are functions that relate an angle in a right angled triangle to the ratio of two of its sides.

From the given,$\sin \theta =\cos \theta$

Divide both sides by $\cos \theta$.

$\Rightarrow$ $\frac{\sin \theta }{\cos \theta }=\frac{\cos \theta }{\cos \theta }$

According to Reciprocal identities of trigonometric function,

$\Rightarrow$ $\tan \theta =\frac{\sin \theta }{\cos \theta }$

$\Rightarrow$ $\tan \theta =1$

$\Rightarrow$ $\tan \theta =\tan 45°$

$\Rightarrow$ $\theta =45°$

$\Rightarrow$ $2{\tan }^2\theta +{\sin }^2\theta -1=2{\tan }^{2}45°+{\sin }^{2}45°-1$

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

$=\frac{3}{2}$

Hence, the value of $2{\tan }^2\theta +{\sin }^2\theta -1$ is $\frac{3}{2}$.