Dgdrxrt

Re-submission of rejected manuscript without informing co-authors

Could Giant Ground Sloths have been a good pack animal for the ancient Mayans?

Can a planet have a different gravitational pull depending on its location in orbit around its sun?

Is there a familial term for apples and pears?

Is it wise to focus on putting odd beats on left when playing double bass drums?

What to wear for invited talk in Canada

How can I fix this gap between bookcases I made?

Is this food a bread or a loaf?

Can I legally use front facing blue light in the UK?

How would photo IDs work for shapeshifters?

What does "enim et" mean?

Need help identifying/translating a plaque in Tangier, Morocco

LWC and complex parameters

I’m planning on buying a laser printer but concerned about the life cycle of toner in the machine

Are objects structures and/or vice versa?

How could a lack of term limits lead to a "dictatorship?"

How to deal with fear of taking dependencies

How to move the player while also allowing forces to affect it

Why is my log file so massive? 22gb. I am running log backups

What is the meaning of "of trouble" in the following sentence?

Extreme, but not acceptable situation and I can't start the work tomorrow morning

Patience, young "Padovan"

Landlord wants to switch my lease to a "Land contract" to "get back at the city"

How is it possible for user's password to be changed after storage was encrypted? (on OS X, Android)

Find the minimum value of $a tan^2 x + b cot^2 x,$ where $text a is greater than b, b is greater than 0.$

more than one minimum value?Find the minimum value of $|sin x+cos x+tan x+cot x+sec x+csc x|$ for real numbers $x$Find the value of $tan^2alpha+cot^2beta$Find the minimum value of $sin^2 theta +cos^2 theta+sec^2 theta+csc^2 theta+tan^2 theta+cot^2 theta$The angle giving minimum valueIf $x + y + z + w=5$ then the minimum value of $x^2 cot (9°) + y^2 cot (27°) + z^2 cot (63°) + w^2 cot (81°)$ is?Find the value of $cos tan^-1 sin cot^-1 (x)$ .Finding the minimum value of $cot^2A + cot^2B+ cot^2C$ where $A$, $B$ and $C$ are angles of a triangle.Find the value of $cot(16)cot(44)+cot(44)cot(76)-cot(76)cot(16)$Find the value of $cot^-121+cot^-113+cot^-1(-8)$

0

$begingroup$

Find the minimum value of $a tan^2 x + b cot^2 x.text a is greater than b, b is greater than 0$.
Closest thing to solution I can come up with is this
$(sqrta tan x + sqrtbcot x )^2 -2 sqrtasqrtb $

algebra-precalculus trigonometry

share|cite|improve this question

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Let $tan(x)=t$, you can find the minimum value of $at^2+b/t^2$ where $t in BbbR$.
  $endgroup$
  – Xin Fu
  Mar 30 at 4:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @XinFu hmm then what ?
  $endgroup$
  – swarnim
  Mar 30 at 4:49
  

  
  
  
  

add a comment | 

0

$begingroup$

Find the minimum value of $a tan^2 x + b cot^2 x.text a is greater than b, b is greater than 0$.
Closest thing to solution I can come up with is this
$(sqrta tan x + sqrtbcot x )^2 -2 sqrtasqrtb $

algebra-precalculus trigonometry

share|cite|improve this question

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Let $tan(x)=t$, you can find the minimum value of $at^2+b/t^2$ where $t in BbbR$.
  $endgroup$
  – Xin Fu
  Mar 30 at 4:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @XinFu hmm then what ?
  $endgroup$
  – swarnim
  Mar 30 at 4:49
  

  
  
  
  

add a comment | 

$begingroup$

Find the minimum value of $a tan^2 x + b cot^2 x.text a is greater than b, b is greater than 0$.
Closest thing to solution I can come up with is this
$(sqrta tan x + sqrtbcot x )^2 -2 sqrtasqrtb $

algebra-precalculus trigonometry

share|cite|improve this question

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

$endgroup$

share|cite|improve this question

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

share|cite|improve this question

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

edited Mar 30 at 4:51

Robert Lewis

48.9k23168

asked Mar 30 at 4:38

swarnimswarnim

938

asked Mar 30 at 4:38

swarnimswarnim

938

3 Answers
3

active

oldest

votes

1

$begingroup$

Let $t=(tan , x)^2$. Note that $at+frac b t =(sqrt at -sqrt frac b t)^2+2sqrt ab geq 2sqrt ab$. Also equality holds when $t=sqrt b/a$. Hence the minimum value is $2sqrt ab$ which is attained when $tan , x=(b/a)^1/4$.

share|cite|improve this answer

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

add a comment | 

1

$begingroup$

The AM-GM inequality says: $atan^2x + bcot^2x ge 2sqrtatan^2xcdot bcot^2x= 2sqrtab$. What does this mean...?

share|cite|improve this answer

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

$endgroup$

add a comment | 

0

$begingroup$

You are almost there:

$(√a|tan x| -√b|cot x|)^2 +$

$2√a√b|tan x||cot x| ge$

$ 2√a√b|tan x cot x| =2√a√b.$

Equality:

$√a|tan x| -√b|cot x| =0.$

$tan ^2 x =√b/√a.$

share|cite|improve this answer

edited Mar 30 at 12:14

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

$endgroup$

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3167933%2ffind-the-minimum-value-of-a-tan2-x-b-cot2-x-where-text-a-is-greater%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

Let $t=(tan , x)^2$. Note that $at+frac b t =(sqrt at -sqrt frac b t)^2+2sqrt ab geq 2sqrt ab$. Also equality holds when $t=sqrt b/a$. Hence the minimum value is $2sqrt ab$ which is attained when $tan , x=(b/a)^1/4$.

share|cite|improve this answer

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

add a comment | 

1

$begingroup$

Let $t=(tan , x)^2$. Note that $at+frac b t =(sqrt at -sqrt frac b t)^2+2sqrt ab geq 2sqrt ab$. Also equality holds when $t=sqrt b/a$. Hence the minimum value is $2sqrt ab$ which is attained when $tan , x=(b/a)^1/4$.

share|cite|improve this answer

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

add a comment | 

$begingroup$

Let $t=(tan , x)^2$. Note that $at+frac b t =(sqrt at -sqrt frac b t)^2+2sqrt ab geq 2sqrt ab$. Also equality holds when $t=sqrt b/a$. Hence the minimum value is $2sqrt ab$ which is attained when $tan , x=(b/a)^1/4$.

share|cite|improve this answer

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

share|cite|improve this answer

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

edited Mar 30 at 5:37

trancelocation

13.7k1829

edited Mar 30 at 5:37

trancelocation

13.7k1829

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

answered Mar 30 at 4:51

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

1

$begingroup$

The AM-GM inequality says: $atan^2x + bcot^2x ge 2sqrtatan^2xcdot bcot^2x= 2sqrtab$. What does this mean...?

share|cite|improve this answer

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

$endgroup$

add a comment | 

1

$begingroup$

The AM-GM inequality says: $atan^2x + bcot^2x ge 2sqrtatan^2xcdot bcot^2x= 2sqrtab$. What does this mean...?

share|cite|improve this answer

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

$endgroup$

add a comment | 

$begingroup$

The AM-GM inequality says: $atan^2x + bcot^2x ge 2sqrtatan^2xcdot bcot^2x= 2sqrtab$. What does this mean...?

share|cite|improve this answer

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

$endgroup$

share|cite|improve this answer

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

answered Mar 30 at 4:51

DeepSeaDeepSea

71.4k54488

0

$begingroup$

You are almost there:

$(√a|tan x| -√b|cot x|)^2 +$

$2√a√b|tan x||cot x| ge$

$ 2√a√b|tan x cot x| =2√a√b.$

Equality:

$√a|tan x| -√b|cot x| =0.$

$tan ^2 x =√b/√a.$

share|cite|improve this answer

edited Mar 30 at 12:14

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

$endgroup$

add a comment | 

0

$begingroup$

You are almost there:

$(√a|tan x| -√b|cot x|)^2 +$

$2√a√b|tan x||cot x| ge$

$ 2√a√b|tan x cot x| =2√a√b.$

Equality:

$√a|tan x| -√b|cot x| =0.$

$tan ^2 x =√b/√a.$

share|cite|improve this answer

edited Mar 30 at 12:14

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

$endgroup$

add a comment | 

$begingroup$

You are almost there:

$(√a|tan x| -√b|cot x|)^2 +$

$2√a√b|tan x||cot x| ge$

$ 2√a√b|tan x cot x| =2√a√b.$

Equality:

$√a|tan x| -√b|cot x| =0.$

$tan ^2 x =√b/√a.$

share|cite|improve this answer

edited Mar 30 at 12:14

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

$endgroup$

share|cite|improve this answer

edited Mar 30 at 12:14

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822

answered Mar 30 at 5:17

Peter SzilasPeter Szilas

11.9k2822