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Prove that :-

[tex] \sf \pmb{ {\rm{( \sin \theta + \sec \theta ) {}^{2} } + (cos \theta + cosec \theta) {}^{2} = (1 + sec \theta cosec \theta) {}^{2}}}[/tex]

Solution :-

[tex] \pmb{ \sf \: L.H.S. =(\sin \theta+\sec \theta)^{2}+(\cos \theta+\operatorname{cosec} \theta)^{2} }[/tex]

[tex] \\ \\ \pmb{\bf =\left(\sin \theta+\frac{1}{\cos \theta}\right)^{2}+\left(\cos \theta+\frac{1}{\sin \theta}\right)^{2}} [/tex]

[tex] \\ \\\pmb{ \bf =(\sin \theta)^{2}+\left(\frac{1}{\cos \theta}\right)^{2}+2(\sin \theta)\left(\frac{1}{\cos \theta}\right)+(\cos \theta)^{2}+\left(\frac{1}{\sin \theta}\right)^{2}+2(\cos \theta)\left(\frac{1}{\sin \theta}\right)} [/tex]

[tex] \\ \\\pmb{ \bf =\sin ^{2} \theta+\frac{1}{\cos ^{2} \theta}+2 \frac{\sin \theta}{\cos \theta}+\cos ^{2} \theta+\frac{1}{\sin \theta}+2 \frac{\cos \theta}{\sin \theta} }[/tex]

[tex] \\ \\\pmb{ \bf =\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+\left(\frac{1}{\cos ^{2} \theta}+\frac{1}{\sin ^{2} \theta}\right)+2\left[\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right]} [/tex]

[tex] \\ \\ \pmb{ \bf =1+\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos ^{2} \theta \sin ^{2} \theta}\right)+2\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos ^{2} \theta \sin \theta}\right) }[/tex]

[tex] \\ \\ \pmb{\bf =\left(1+\frac{1}{\cos ^{2} \theta \sin ^{2} \theta}+\frac{2 \times 1}{\cos \theta \sin \theta}\right)} [/tex]

[tex] \\ \\ \pmb{\bf =1+\sec ^{2} \theta \operatorname{cosec}^{2} \theta+2 \sec \theta \operatorname{cosec} \theta} [/tex]

[tex] \\ \\ \pmb{ \bf=(1)^{2}+(\sec \theta \operatorname{cosec} \theta)^{2}+2 \times 1 \times(\sec \theta \operatorname{cosec} \theta)} [/tex]

[tex] \\\pmb{ \bf =[1+\sec \theta \operatorname{cosec} \theta]^{2}= R.HS.}[/tex]

[tex]\rule{500pt}{0.1pt}[/tex]

ησtε :-

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