(1) If A + B + C = 180°, show that

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

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

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

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

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

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

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

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

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

(2x/1 − x^{2}) + (2y/1 − y^{2}) + (2z/1 − z^{2}) = (2x/1 − x^{2}) (2y/1 − y^{2}) (2z/1 − z^{2}) Answer

(4) If A + B + C = π/2, show the next

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

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

Answer

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

(i) cos^{2} B + cos^{2} C = 1

(ii) sin^{2} B + sin^{2} C = 1

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

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