The minimum value of sin^{2} theta + cos^{2} | vedclass

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(D) Let the expression be $E = \sin^{2} 	heta + \cos^{2} 	heta + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	he

Vedclass · Accepted Answer

$7$

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(D) Let the expression be $E = \sin^{2} 	heta + \cos^{2} 	heta + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	heta$.Using the identity $\sin^{2} 	heta + \cos^{2} 	heta = 1$,we have $E = 1 + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	heta$.Using $\sec^{2} 	heta = 1 + 	an^{2} 	heta$ and $\operatorname{cosec}^{2} 	heta = 1 + \cot^{2} 	heta$,we get:$E = 1 + (1 + 	an^{2} 	heta) + (1 + \cot^{2} 	heta) + 	an^{2} 	heta + \cot^{2} 	heta$.$E = 3 + 2(	an^{2} 	heta + \cot^{2} 	heta)$.By the Arithmetic Mean-Geometric Mean inequality,$	an^{2} 	heta + \cot^{2} 	heta \ge 2 \sqrt{	an^{2} 	heta \cdot \cot^{2} 	heta} = 2(1) = 2$.Therefore,the minimum value of $E = 3 + 2(2) = 3 + 4 = 7$.

The minimum value of $\sin^{2} \theta + \cos^{2} \theta + \sec^{2} \theta + \operatorname{cosec}^{2} \theta + \tan^{2} \theta + \cot^{2} \theta$ is

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