**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.

Here we are going to see some example problems to show how to solve conditional trigonometric identities problems.

**Example 1 :**

If A + B + C = 2s, then prove that sin(s − A) sin(s − B) + sins sin(s − C) = sin A sin B.

**Solution :**

A + B + C = 2s

L.H.S :

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

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) - cos C] ------(1)

sins sin(s − C) = (2/2) sins sin(s − C)

= (1/2)(2 sins sin (s - C))

= (1/2)[cos C - cos (2s - C)] ------(2)

(1) + (2)

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

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

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

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

= (1/2) [-2 sin B sin (-A)]

= sin A sin B

Hence proved

**Example 2 :**

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

(2x/1 − x^{2}) + (2y/1 − y^{2}) + (2z/1 − z^{2}) = (2x/1 − x^{2}) (2y/1 − y^{2}) (2z/1 − z^{2})

**Solution :**

x + y + z = xyz

Let x = tan A, y = tan B and z = tan C

tan A + tan B + tan C = tan A tan B tan C

tan A + tan B = tan A tan B tan C - tan C

tan A + tan B = tan C (tan A tan B - 1)

tan A + tan B = -tan C (1 - tan A tan B)

(tan A + tan B) / (1 - tan A tan B) = - tan C

tan (A + B) = tan (-C)

A + B = -C

A + B + C = 0

2A + 2B + 2C = 0

tan (2A + 2B) = tan (- 2C)

(tan 2A + tan 2B)/(1 - tan 2A tan 2B) = - tan 2C

(tan 2A + tan 2B) = - tan 2C(1 - tan 2A tan 2B)

tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C-----(1)

tan 2A = 2 tan A / 1 - tan^{2}A = 2x/(1 -x^{2})

tan 2B = 2 tan B / 1 - tan^{2}B = 2y/(1 -y^{2})

tan 2C = 2 tan C / 1 - tan^{2}C = 2z/(1 -z^{2})

By applying those values in (1), we get

2x/(1 -x^{2}) + 2y/(1 -y^{2}) + 2z/(1 - z^{2}) = 2x/(1 -x^{2}) 2y/(1 -y^{2}) 2z/(1 - z^{2})

Hence proved.

After having gone through the stuff given above, we hope that the students would have understood how to solve problems in trigonometry using conditional identities.

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