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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

$7$

Vedclass · Answer

(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

Similar Questions

If $\operatorname{cosec} \theta - \sin \theta = m$ and $\sec \theta - \cos \theta = n$,then $(m^2 n)^{2/3} + (m n^2)^{2/3} = $

$ABC$ is a triangle. If $\sin \left(\frac{A+B}{2}\right) = \frac{\sqrt{3}}{2}$,then the value of $\sin \frac{C}{2}$ is

If $\sin \theta = \frac{1}{2} \left( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} \right)$,where $x, y \in R - \{0\}$. Then:

The extreme values of $1 + 4 \sin \theta + 3 \cos \theta$ are:

$\cos \frac{\pi }{5} \cos \frac{2\pi }{5} \cos \frac{4\pi }{5} \cos \frac{8\pi }{5} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module