CONDITIONAL IDENTITIES TRIGONOMETRIC EXAMPLES

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 examples to show how to solve conditional trigonometric identities problems.

Example 1 :

If A + B + C  =  180°, prove that 

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

Solution :

A + B + C  =  180

A + B  =  180 - C

A/2 + B/2  =  90 - C/2

tan(A/2 + B/2)  =  tan[90 - (C/2)]

tan(A/2 + B/2)  =  cot(C/2) -----(1)

Formula :  

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

Use the above formula on the left side of (1). 

(1)----->

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

Take reciprocal on both sides

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

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

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

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

Example 2 :

If A + B + C  =  180°, prove that 

sinA + sinB + sinC  =  4cos A/2 cos B/2 cos C/2

Solution :

sinA + sinB + sinC :

=  2sin(A + B)/2cos(A - B)/2 + sinC

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

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

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

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

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

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

=  2 sin C/2 [2 cos A/2 cos B/2]

=  4 cos A/2 cos B/2 sin C/2

Example 3 :

If A + B + C  =  180°, prove that 

sin(B + C - A) + sin(C + A − B) + sin(A + B - C) = 4sinAsinB sinC.

Solution :

A + B + C  =  180

sin(B + C - A) + sin(C + A - B) + sin(A + B - C) :

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

=  sin(180 - 2A) + sin(180 - 2B) + sin(180 - 2C)

=  sin 2A + sin 2B + sin 2C

=  2sin(A + B)cos(A - B) + sin2C

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

=  2sinCcos(A - B) + 2sinCcosC

=  2sinC[cos(A - B) + cosC]

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

=  2sinC[cos(A - B) - cos(A + B)]

=  2sinC[-2sinAsin(-B)]

=  4sinCsinAsinB

=  4sinAsinBsinC

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