Question

${\left(\sin \theta +\cos ec\theta \right)}^2+{\left(\cos \theta +\sec \theta \right)}^2=7+{\tan }^2\theta +{\cot }^2\theta$

Answer 1

$(\sin \theta +cosec\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$

Given, L.H.S$=(\sin \theta +cosec\theta {)}^2+(\cos \theta +\sec \theta {)}^2$

$\Rightarrow$ L.H.S$={\sin }^2\theta +cose{c}^2\theta +2\sin \theta \cdot cosec\theta +{\cos }^2\theta +{\sec }^2\theta +2\cos \theta \cdot \sec \theta$ $(\because (a+b{)}^2=a^2+b^2+2ab)$

$\Rightarrow$ L.H.S$={\sin }^2\theta +{\cos }^2\theta +{\sec }^2\theta +cose{c}^2\theta +2\cdot \sin \theta \cdot \frac{1}{\sin \theta }+2\cdot \cos \theta \cdot \frac{1}{\cos \theta }$ ($\because$ Using $cosec\theta =\frac{1}{\sin \theta }+\sec \theta =\frac{1}{\cos \theta }$)

$\Rightarrow$ L.H.S.$=1+cose{c}^2\theta +{\sec }^2\theta +2+2$ ($\because$ Using identify ${\sin }^2\theta +{\cos }^2\theta =1$)

$\Rightarrow$ L.H.S$=5+(1+{\cot }^2\theta )+(1+{\tan }^2\theta )$ $(\because cose{c}^2\theta =1+{\cot }^2\theta ,{\sec }^2\theta =1+{\tan }^2\theta )$

$\Rightarrow$ L.H.S.$=7+{\tan }^2\theta +{\cot }^2\theta$

$\Rightarrow$ L.H.S$=$R.H.S

$\therefore (\sin \theta +cosec\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$.