Question

If $\csc \theta -\cot \theta =\surd 2.\cot \theta$, then prove that $\csc \theta +\cot \theta =\surd 2.\csc \theta$

Answer 1

Given:- $\csc \theta -\cot \theta =\sqrt{2}\cot \theta$

To prove:- $\csc \theta +\cot \theta =\sqrt{2}\csc \theta$

Proof:-

$\csc \theta -\cot \theta =\sqrt{2}\cot \theta$

Squaring both sides, we get

${\csc }^{2}t+{\cot }^{2}t-2\csc \theta \cot \theta =2{\cot }^2\theta$

${\csc }^2\theta +{\cot }^2\theta +2\csc \theta \cot \theta =2{\cot }^2\theta +4\csc \theta \cot \theta$

${(\csc \theta +\cot \theta )}^2={(\sqrt{2}\csc \theta +\sqrt{2}\cot \theta )}^2-2{\csc }^2\theta$

$\Rightarrow 2\csc \theta =2{(\csc \theta +\cot \theta )}^2-{(\csc \theta +\cot \theta )}^2$

$\Rightarrow \csc \theta +\cot \theta =\sqrt{2}\csc \theta$

Hence proved.