Prove that 3cot2-tan2-8cot -tan -10- 0 for real - n -n -2 where n - Z-

3(^2θ+tan^2θ) 8( θ+tan θ)+10≥ 0 3(cos ^ 4 θ +sin ^ 4 θ ) 8sinθ nbsp;cosθ nbsp; +10cos ^ 2 θ nbsp;sin ^ 2 θ nbsp;≥ 0 3 8sinθ nbsp;cosθ ...

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Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

Prove that ...

Question

Prove that $3 (c o t^{2} θ + t a n^{2} θ) - 8 (c o t θ + t a n θ) + 10 \geq 0$ for real $θ \neq n π, n π + \frac{π}{2} w h e r e n \in Z$ .

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Solution

$3 ({cot}^{2} θ + {tan}^{2} θ) - 8 (cot θ + tan θ) + 10 \geq 0$
$3 ({cos}^{4} θ + {sin}^{4} θ) - 8 sin θ cos θ + 10 {cos}^{2} θ {sin}^{2} θ \geq 0$
$3 - 8 sin θ cos θ + 4 {cos}^{2} θ {sin}^{2} θ \geq 0$
${sin}^{2} 2 θ - 4 sin 2 θ + 3 \geq 0$
$\Rightarrow (sin 2 θ - 1) (sin 2 θ - 3) \geq 0$

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Similar questions

Q. The general solution(s) of

θ

satisfying the equation

{tan}^{2} θ + sec 2 θ = 1

can be (where

n \in Z

)

Q. If

{tan}^{3} θ = tan ϕ

and

tan 2 θ = 2 tan α,

prove that -

θ + ϕ = n π + α

Consider the system of linear equations in x, y and z ;
(sin 3 $θ$ ) x - y + z = 0 ......(i)
(cos 2 $θ$ )x + 4y + 3z = 0 ......(ii)
2x + 7y+ 7z = 0 ......(iii)

The value of $θ$ for which the system has nontrivial solution is

Q. If

| 2 sin θ - cosec θ | \geq 1

, then complete set of values of

θ

is
(where

n \in Z

)

Q. Solution of

3 ({sec}^{2} θ + {tan}^{2} θ) = 5

θ = n π \pm \frac{π}{m}

. Find

m

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Prove that 3(cot2θ+tan2θ)−8(cotθ+tanθ)+10≥0 for real θ≠nπ,nπ+π2wheren∈Z.

3(cot2θ+tan2θ)−8(cotθ+tanθ)+10≥03(cos4θ+sin4θ)−8sinθcosθ+10cos2θsin2θ≥03−8sinθcosθ+4cos2θsin2θ≥0sin22θ−4sin2θ+3≥0⇒(sin2θ−1)(sin2θ−3)≥0

Prove that $3 (c o t^{2} θ + t a n^{2} θ) - 8 (c o t θ + t a n θ) + 10 \geq 0$ for real $θ \neq n π, n π + \frac{π}{2} w h e r e n \in Z$ .