CONDITIONAL TRIGONOMETRIC IDENTITIES WORKSHEET

(1)  If A + B + C = 180°, prove that 

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

Solution

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

Solution

(iii) sin² A + sin² B + sin² C = 2 + 2cosAcosB cosC

Solution

(iv)  sin² A + sin² B − sin² C = 2 sin A sin B cos C

Solution

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

Solution

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

Solution

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

(2)  If A + B + C = 2s, then prove that sin(s − A) sin(s − B) + sins sin(s − C) = sin A sin B.          Solution

(3)  If x + y + z = xyz, then prove that

(2x/1 − x²) + (2y/1 − y²) + (2z/1 − z²) = (2x/1 − x²) (2y/1 − y²) (2z/1 − z²)    Solution

(4)  If A + B + C = π/2, prove the following

(i) sin 2A + sin2B + sin2C = 4cosAcosB cosC      Solution

(ii) cos 2A + cos2B + cos2C = 1 + 4sinAsinB cosC 

Solution 

(5)  If triangle ABC is a right triangle and if ∠A = π/2, then prove that

(i) cos² B + cos² C = 1

(ii) sin² B + sin² C = 1

(iii) cosB − cosC = −1 + 2 √2 cos B/2 sin C/2      Solution

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