PROBLEMS INVOLVING CONDITIONAL IDENTITIES IN TRIGONOMETRY

Trigonometric identities are true for all admissible values of the angle involved. There are some trigonometric identities which satisfy the given additional conditions. Such identities are called conditional trigonometric identities.

Problem 1 :

If A + B + C = 2s, then prove that

sin(s − A) sin(s − B) + sins sin(s − C)  =  sinA sinB

Solution :

Given : A + B + C  =  2s.

sin(s − A) sin(s − B) :

=  (2/2)sin(s − A) sin(s − B)

=  (1/2)[cos(s - A - s + B) - cos(s - A + s - B)]

=  (1/2)[cos(B - A) - cos(2s - A - B)]

=  (1/2)[cos(B - A) - cos(A + B + C - A - B)]

=  (1/2)[cos(B - A) - cosC]

sins sin(s − C) :

=  (2/2)sins sin(s − C)

=  (1/2)[2 sins sin(s - C)]

=  (1/2)[cosC - cos(2s - C)]

sin(s − A) sin(s − B) + sins sin(s − C) :

=  (1/2)[cos(B - A) - cosC] + (1/2)[cosC - cos(2s - C)]

=  (1/2)[cos(B - A) - cosC + cosC - cos(2s - C)]

=  (1/2)[cos(B - A) - cos(A + B + C - C)]

=  (1/2)[cos(B - A) - cos(A + B)]

=  (1/2)[-2sinB sin(-A)]

=  sinA sin B

Hence proved

Problem 2 :

If x + y + z = xyz, then prove that

(2x/1 − x2) + (2y/1 − y2) + (2z/1 − z2)

=  (2x/1 − x2) (2y/1 − y2) (2z/1 − z2)

Solution :

x + y + z  =  xyz

Let x = tanA, y = tanB and z = tanC.

Then,

x + y + z  =  xyz

tanA + tanB + tanC  =  tanA tanB tanC

tanA + tanB  =  tanA tanB tanC - tanC

tanA + tanB  =  tanC(tanAtanB - 1)

tanA + tanB  =  -tanC(1 - tanAtanB)

(tanA + tanB) / (1 - tanAtanB)  =  - tanC

tan(A + B)  =  tan(-C)

A + B  =  -C

Multiply each side by 2.

2A + 2B  =  -2C

tan(2A + 2B) =  tan(- 2C)

(tan2A + tan2B)/(1 - tan2Atan2B)  =  -tan2C

(tan2A + tan2B)  =  -tan2C(1 - tan2Atan2B)

tan2A + tan2B + tan2C  =  tan2Atan2Btan2C -----(1)

tan2A  =  2tanA / 1 - tan2A  =  2x/(1 - x2)

tan2B  =  2tanB / 1 - tan2B  =  2y/(1 - y2)

tan2C  =  2tanC / 1 - tan2C  =  2z/(1 - z2)

Substitute these in (1).

2x/(1 -x2) + 2y/(1 -y2) + 2z/(1 - z2

=  2x/(1 -x2)  2y/(1 -y2) 2z/(1 - z2)

Hence proved. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

Recent Articles 1. Multiplicative Inverse Worksheet

Jan 19, 22 09:34 AM

Multiplicative Inverse Worksheet

2. Multiplicative Inverse

Jan 19, 22 09:25 AM

Multiplicative Inverse - Concept - Examples