Question

The value of $\cot \theta \tan ({90}^o-\theta )-\sec ({90}^o-\theta )\text{cosec}\theta +{\sin }^2{25}^o+{\sin }^2{65}^o+\sqrt{3}\left(\tan 5^o.\tan {45}^o.\tan {85}^o\right)$ is :

• A. $1$
• B. $\sqrt{2}$
• C. $\sqrt{3}$
• D. $-1$

Answer 1

The correct option is C $\sqrt{3}$

$\cot \theta \tan ({90}^o-\theta )-\sec ({90}^o-\theta )\text{cosec}\theta +{\sin }^2{25}^o+{\sin }^2{65}^o+\sqrt{3}\left(\tan 5^o\tan {45}^o\tan {85}^o\right)$

$=\cot \theta .\cot \theta -\text{cosec}\theta .\text{cosec}\theta +{\sin }^2{25}^o+{\sin }^2({90}^o-{25}^o)+\sqrt{3}\left\{\tan 5^o\times 1\times \tan ({90}^o-5^o)\right\}$

$={cot}^2\theta -{\text{cosec}}^2\theta +{\sin }^2{25}^o+{cos}^2{25}^o+\sqrt{3}\left\{\tan 5^o\cot 5^o\right\}$

$={\text{cosec}}^2\theta -1-{\text{cosec}}^2\theta +1+\sqrt{3}\left(1\right)$

$[\because {\cot }^2\theta ={\text{cosec}}^2\theta -1;{\sin }^2{25}^o+{\cos }^2{25}^o=1and\tan 5^o\cot 5^o=1]$

$=-1+1+\sqrt{3}=\sqrt{3}$