# CONDITIONAL TRIGONOMETRIC IDENTITIES PROBLEMS

## About "Conditional Trigonometric Identities Problems"

Conditional Trigonometric Identities Problems :

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.

## Conditional Identities Solved Problems

Question 1 :

If A + B + C = 180°, prove that

(i) sin 2A + sin2B + sin2C = 4 sin A sin B sin C

Solution :

L.H.S

=  sin 2A + sin2B + sin2C

Let us use the formula for sin C + sin D

=  2 sin (2A + 2B)/2 cos (2A - 2B)/2 + sin 2C

=  2 sin (A + B) cos (A - B) + sin 2C  -----(1)

A + B + C = 180°

A + B  =  180 - C

sin (A + B)  =  sin (180 - C)  =  sin C

Applying sin (A + B)  =  sin C in (1), we get

=  2 sin C cos (A - B) + 2 sin C cos C

=  2 sin C [cos (A - B) + cos C]

=  2 sin C [cos (A - B) + cos (180 - (A + B))]

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

cos C - cos D  =  -2 sin (C + D)/2 sin (C - D)/2

Here C  =  A - B and D =  A  + B

C  + D  =  A - B + A + B  =  2A

C  - D  =  A - B - A - B  =  -2B

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

=  2 sin C [2 sin A sin B]

=  4 sin A sin B sin C

(ii)  cos A + cos B − cos C = −1 + 4cos(A/2)cos(B/2)sin(C/2)

Solution :

L.H.S :

=  cos A + cos B − cos C

=  2 cos (A + B)/2 cos (A - B)/2 − cos C

=  2 cos (180 - C)/2 cos (A - B)/2 − cos C

=  2 cos (90 - (C/2)) cos (A - B)/2 − cos C

=  2 sin (C/2) cos (A - B)/2 − (1 - 2 sin2 (C/2))

=  2 sin (C/2) cos (A - B)/2 − 1 + 2 sin2 (C/2)

=  -1 + 2 sin (C/2) cos (A - B)/2 + 2 sin2 (C/2)

=  -1 + 2 sin(C/2) [cos (A - B)/2 + sin (180 - (A + B))/2]

=  -1 + 2 sin(C/2) [cos (A - B)/2 + sin (90 - (A + B)/2)]

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

Applying the formula for cos C + cos D

Here C  =  (A - B) / 2 and D  =  (A + B)/2

C + D  =  2A/2  =  A

C - D  =  -2B/2  =  -B

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

=  -1 + 4 sin(C/2) [cos (A/2) cos (B/2)]

=  -1 + 4 cos (A/2) cos (B/2) sin(C/2)

After having gone through the stuff given above, we hope that the students would have understood, "Conditional Trigonometric Identities Problems"

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