Question

सिद्ध कीजिए। 

यदि ${\displaystyle \tan \theta +\frac{1}{\tan \theta }=2}$ तो सिद्ध कीजिए कि ${\displaystyle {\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2}$

Answer 1

${\displaystyle \tan \theta +\frac{1}{\tan \theta }=2}$ .........(दिया है |) ..............(1)

दोनों पक्षों का वर्ग करने पर,

${\displaystyle {(\tan \theta +\frac{1}{\tan \theta })}^2={\left(2\right)}^2}$

∴ ${\displaystyle {\tan }^2\theta +2\left(\tan \theta \right)\left(\frac{1}{\tan \theta }\right)+\frac{1}{{\tan }^2\theta }=4}$ .............[∵ (a + b)² = a² + 2ab + b²] 

∴ ${\displaystyle {\tan }^2\theta +2+\frac{1}{{\tan }^2\theta }=4}$

∴ ${\displaystyle {\tan }^2\theta +\frac{1}{{\tan }^2\theta }=4-2}$

∴ ${\displaystyle {\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2}$.