Conditional Identities

You have studied identities like (x +y)² = x² +y² +2xy, which hold for all values (of x and y). Conditional identity is an identity which holds if variables satisfy a given condition.

Method of solving conditional identities

  1. If the identity involves sines and cosines of angles, then sums/differences should be converted into products using A-B formulae and then simplification should be done, by using C-D formulae or other relevant formulae.
  2. If the identity involves squares of sines or cosines, first the squares should be changed into cosines of double angles by using the formulae
       cos²A= (1 +cos2A)/2 and sin²A = (1 -cos2A)/2
  3. If tangents or cotangents are involved, express the sum of two angles in terms of third angle, (from given relation) and then take tangents of both sides and expand, and simplify.

Illustrative Examples

Example

If A, B, C are angles of a triangle, prove that
     sin 2A +sin 2C = 4 sin A sin B sin C

Solution

Since A, B, C are angles of a triangle, we have A +B +C = = 180°.
L.H.S. = (sin 2A +sin 2B) +sin 2C
= 2 sin [(2A +2B)/2] cos [(2A -2B)/2] + sin 2C
= 2 sin (A +B) cos (A -B) + 2 sin C cos C
= 2 sin ( -C) cos (A -B) + 2 sin C cos ( -(A +B))
= 2 sin C cos (A -B) -2 sin C cos (A +B)
= 2 sin C [cos (A -B) -cos (A +B)]
= 2 sin C (2 sin A sin B) = 4 sin A sin B sin C = R.H.S.

Example

If A +B +C = , prove that tan A +tan B + tan C = tan A tan B tan C

Solution

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

Exercise

  1. If A +B +C = , prove that
    (i) sin 2A +sin 2B -sin 2C = 4 cos A cos B sin C
    (ii) sin 2A -sin 2B +sin 2C = 4 cos A sin B cos C
    (iii) cos 2A +cos 2B +cos 2C = -1 -4 cos A cos B cos C
    (iv) cos 2A -cos 2B +cos 2C = -1 -4 sin A cos B sin C.
  2. If A, B, C are angles of a triangle, prove that
    (i) cos²A + cos²B +cos²C = 1 -2 cos A cos B cos C
    (ii) sin²A -sin²B + sin²C = 2 sin A cos B sin C
  3. If A +B +C = , prove that
    (i) tan 2A +tan 2B +tan 2C = tan 2A tan 2B tan 2C
    (ii) tan A/2 tan B/2 + tan B/2 tan C/2 +tan C/2 tan A/2 = 1
  4. If A +B +C = , prove that
    sin A/2 +sin B/2 +sin C/2 = 1 + 4 sin ( -A)/4 sin ( -B)/4 sin ( -C)/4
  5. If A +B +C = 90°, prove that
    (i) sin 2A +sin 2B +sin 2C = 4 cos A cos B cos C
    (ii) cos²A + cos²B +cos²C = 2 +2 sin A sin B sin C
    (iii) tan A tan B +tan B tan C +tan C tan A = 1.