**Conditional Identities Solved 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.

**Question 1 :**

If A + B + C = 180°, prove that

(iii) sin^{2} A + sin^{2} B + sin^{2} C = 2 + 2cosAcosB cosC

**Solution :**

L.H.S

= sin^{2} A + sin^{2} B + sin^{2} C

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

= (3/2) - (1/2)[cos 2A + cos 2B + cos 2C]

= (3/2) - (1/2)[2 cos (A + B) cos (A - B) + 2cos^{2}C - 1]

= (3/2) - (1/2)[2 cos (180-C) cos (A - B) + 2cos^{2}C - 1]

= (3/2) - (1/2)[-2 cos C cos (A - B) + 2cos^{2}C - 1]

= (3/2) - [- cos C cos (A - B) + cos^{2}C] + (1/2)

= (3/2) - [- cos C cos (A - B) + cos^{2}C] + (1/2)

= (3/2) + (1/2) + cos C [cos (A - B) - cos C]

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

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

Now let us use the formula for cos C - cos D

= 2 + cos C [2 cos A cos B]

= 2 + 2 cos A cos B cos C

(iv) sin^{2} A + sin^{2} B − sin^{2} C = 2 sin A sin B cos C

**Solution :**

L.H.S

= sin^{2} A + sin^{2} B − sin^{2} C

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

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

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

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

= (1/2) - (1/2)[-2 cos C cos (A - B) - (2cos^{2}C - 1)]

= (1/2) + cos C cos (A - B) + cos^{2}C - (1/2)

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

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

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

= cos C [2 sin A sin B]

= 2 sin A sin B cos C

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

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