Home
Class 12
MATHS
If A+B+C=pi, prove that tan^2A/2+tan^2B/...

If `A+B+C=pi,` prove that `tan^2A/2+tan^2B/2+tan^2C/2geq1.`

Text Solution

Verified by Experts

`(tan""(A)/(2)-tan""(B)/(2))^(2)+(tan""(B)/(2)-tan""(C)/(2))^(2)+(tan""(C)/(2)-tan""(A)/(2))^(2)ge0`
`therefore 2(tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2))`
`-2(tan""(A)/(2)tan""(B)/(2)+tan""(C)/(2)tan""(A)/(2)tan""(C)/(2))ge0`
we know that in `triangle ABC`,
`tan ""(A)/(2)tan""(B)/(2)+tan""(B)/(2)tan""(C)/(2)+tan""(A)/(2)tan ""(C)/(2)=1`
`therefore 2(tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2))-2(1)ge0`
`therefore tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1`
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.1|11 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.2|7 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise SINGLE CORRECT ANSWER TYPE|38 Videos

  • TRIGONOMETRIC RATIOS FOR COMPOUND, MULTIPLE, SUB-MULTIPLE ANGLES, AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Multiple Correct Answers Type|6 Videos

Similar Questions

Explore conceptually related problems

In a triangle ABC, prove that: tan^2, A/2+tan^2, B/2+tan^2, C/2ge1

If A+B+C=pi, prove that tan^(2)(A)/(2)+tan^(2)(B)/(2)+tan^(2)(C)/(2)>=1

If A+B+C=pi , prove that : tan( A/2) tan (B/2) + tan (B/2 )tan (C/2)+ tan( C/2) tan (A/2) =1

If A+B+C=pi, prove that (tan A)/(tan B tan C)+(tan B)/(tan Atan C)+(tan C)/(tan A tan B)=tan A+tan B+tan C-2cot A-2cot B-2cot C

If A+B+C= pi/2 , prove that tan A tan B+tan B tan C+tan C tan A=1

If A+B+C=180^(@) , prove that: "tan"2A+ "tan"2B+"tan" 2C="tan" 2A "tan"2B tan 2C

If A+B+C=pi, show that : tan.(A)/(2)tan.(B)/(2)+tan.(B)/(2)tan.(C)/(2)+tan.(C)/(2)tan.(A)/(2)=1 Hence deduce that : cot.(A)/(2)+cot.(B)/(2)+.cot.(C)/(2)=cot.(A)/(2).cot.(B)/(2)tan.(C)/(2) .

If A+B+C=pi then prove that sum((tan A)/(tan B tan C))=sum tan A-2sum cot A

In DeltaABC , prove that: tan2A + tan2B+tan2C=tan2Atan2Btan2C