## Note on Conditional Trigonometric Identities

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Identities which are true under some given conditions are termed as conditional identities.

In this section, we will deal some trigonometric identities which are bound to the condition of the sum of the angles of a triangle i.e. A + B + C =π

Properties of supplementary and complementary angles

(i)  Since   A + B + C = π

Then,   A + B = π - C, B + C = π - A and A + C = π - B

Now,   sin(A + B) = sin(π - C) = sin C

sin(B + C) = sin( π - A) = sin A

sin(A + C) = sin(π -B) = sin B

Again, cos(A + B) = cos(π - ) = -cos C

cos(B + C) = cos(π - A) = -cos A

cos(A + C) = cos(π - B) = -cos B

Also,   tan(A + B) = tan(π- C) = -tan B

tan(B + C) = tan(π- A) = -tan A

tan(A + C) = tan(π - B) = -tan B

(ii) Since    A + B + C = π

Then,      $$\frac{A}{2}$$ + $$\frac{B}{2}$$ + $$\frac{C}{2}$$ = $$\frac{π}{2}$$. So, $$\frac{A + B}{2}$$ = $$\frac{π}{2}$$ - $$\frac{C}{2}$$, $$\frac{B + C}{2}$$ = $$\frac{π}{2}$$ - $$\frac{A}{2}$$ and $$\frac{A + C}{2}$$ = $$\frac{π}{2}$$ - $$\frac{B}{2}$$

Now,       sin($$\frac{A + B}{2}$$) = sin($$\frac{π}{2}$$ - $$\frac{C}{2}$$) = cos $$\frac{C}{2}$$

sin($$\frac{B + C}{2}$$) = sin($$\frac{π}{2}$$ - $$\frac{A}{2}$$) = cos $$\frac{A}{2}$$

sin($$\frac{A + C}{2}$$) = sin($$\frac{π}{2}$$ - $$\frac{B}{2}$$) = cos $$\frac{B}{2}$$

Again,      cos($$\frac{A + B}{2}$$) = cos($$\frac{π}{2}$$ - $$\frac{C}{2}$$) = sin $$\frac{C}{2}$$

cos($$\frac{A + C}{2}$$) = cos($$\frac{π}{2}$$ - $$\frac{B}{2}$$) = sin $$\frac{B}{2}$$

cos($$\frac{B + C}{2}$$) = cos($$\frac{π}{2}$$ - $$\frac{A}{2}$$) = sin $$\frac{A}{2}$$

Also,       tan($$\frac{A +B}{2}$$) = tan($$\frac{π}{2}$$ - $$\frac{C}{2}$$) = cot $$\frac{C}{2}$$

tan($$\frac{A + C}{2}$$) = tan($$\frac{π}{2}$$ - $$\frac{B}{2}$$) = cot $$\frac{B}{2}$$

tan($$\frac{B + C}{2}$$) = tan($$\frac{π}{2}$$ - $$\frac{B}{2}$$) = cot $$\frac{A}{2}$$

Conditional Trigonometric Identities

• Properties of supplementary and complementary angles
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