InterviewSolution
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InterviewSolution
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What is the minimum value of sin2 θ + cos2 θ + sec2 θ + cosec2 θ + tan2 θ + cot2 θ1). 12). 33). 54). 7 |
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Answer» We know that sin2 θ + cos2 θ = 1 Therefore, (sin2 θ + cos2 θ) + sec2 θ + cosec2 θ + tan2 θ + cot2 θ ⇒ 1 + sec2 θ + cosec2 θ + tan2 θ + cot2 θ Using A.M ≥ G.M logic for tan2 θ + cot2 θ we get, ⇒ 1 + 2 + $({\sec ^2}{\rm{\theta \;}} + {\rm{\;cose}}{{\rm{c}}^2}{\rm{\;\theta }})$ Changing into sin and cos values (Because we know maximum and minimum values of Sin θ, Cos θ and by using simple identities we can convert all trigonometric functions into equation with SINE and Cosine.) $(\Rightarrow {\rm{\;}}1 + {\rm{\;}}2 + \left( {\frac{1}{{{{\cos }^2}{\rm{\theta }}}}} \right) + \left( {\frac{1}{{{{\sin }^2}{\rm{\theta }}}}} \right))$ Solving taking L.C.M $( \Rightarrow 1 + 2 + \frac{{{{\sin }^2}{\rm{\theta \;}} + {\rm{\;}}{{\cos }^2}{\rm{\theta }}}}{{{{\sin }^2}{\rm{\theta \;}}.{{\cos }^2}{\rm{\theta }}}})$ ------ Equation (1) But we already know two things Min. value of (sin θ cos θ)n = (½)n Apply them into Equation (1), and we get ⇒ 1 + 2 + (sin2 θ + cos2 θ)/( sin2 θ . cos2 θ) = 1 + 2 + (1/1/4) = 1 + 2 + 4 = 7 |
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