Home
Class 12
MATHS
In any triangle A B C , prove that follo...

In any triangle `A B C` , prove that following : `c/(a+b)=(1-tan(A/2)tan(B/2))/(1+tan(A/2)tan(B/2))`

Text Solution

Verified by Experts

`(c)/(a + b) = (2R sin C)/(2R (sin A + sin B)) = (sin (A + B))/(sin A + sin B)`
`= (2 sin.(A + B)/(2) cos.(A + B)/(2))/(2 sin.(1)/(2) (A + B) cos.(1)/(2) (A - B))`
`= (cos.(1)/(2) (A + B))/(cos.(1)/(2) (A - B))`
`= (cos.(1)/(2) A cos.(1)/(2) B - sin.(1)/(2) A sin.(1)/(2) B)/(cos.(1)/(2) A cos.(1)/(2) B + sin.(1)/(2) A sin.(1)/(2)B)`
`= (1 - tan.(1)/(2) A tan.(1)/(2) B)/(1 + tan.(1)/(2) A tan.(1)/(2) B)` [dividing numerator and denominator by `cos.(1)/(2) A cos.(1)/(2) B`]
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.2|8 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.3|3 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Illustration|86 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE )|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1119 Videos

Similar Questions

Explore conceptually related problems

IF A+B+C=pi ,then prove that tan (A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)=1

In an acute angled triangle ABC, prove that tan^(2)a+tan^(2)B+tan^(2) C ge9 .

In triangle ABC prove that tan((A+B)/2)=cot. C/2

Prove that 1-tan(A/2)tan(B/2)=(2c)/(a+b+c) .

In any triangle ABC, the value of (b+c) tan ""A/2 tan ""(B-C)/(2)+ (c+a) tan ""B/2 tan ""(C-A)/(2) + (a+b) tan ""C/2 tan ""(A-B)/(2) is-

In triangle A B C , if |[1, 1, 1],[cot(A/2),cot(B/2),cot(C/2)],[tan(B/2)+tan(C/2),tan(C/2)+tan(A/2),tan(A/2) +tan(B/2)]| =0 then the triangle must be a. equilateral b. isosceles c. obtuse angled d. none of these

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

If A,B,C are the angles of a triangle, show that, (iv) (tan(B + C) + tan (C + A) + tan (A + B))/(tan (pi - A) + tan(2pi - B) + tan (3pi - C)) = 1

In any triangle ABC, if a,b,c are in A.P. then tan ""A/2, tan ""B/2, tan ""C/2 are in -

Prove that (r+r1)tan((B-C)/2)+(r+r2)tan((C-A)/2)+(r+r3)tan((A-B)/2)=0