IN A TRIANGLE ABC , THE ANGLE A, B, C ARE IN A P SHOW THAT 2 COS(A-C )/2 - Maths - Trigonometric Functions

Question

IN A TRIANGLE ABC , THE ANGLE A, B, C ARE IN A P SHOW THAT 2
COS(A-C )/2 = (A+C)/ SQUARE ROOT OF (A SQUARE -AC +C SQUARE)

Lalit Mehra · Accepted Answer

In △ABC, A, B, C are in AP
⇒2B=A+CA+B+C=180°⇒3B=180°⇒B=60°Now,cosB=a2+c2-b22ac⇒cos60°=12=a2+c2-b22ac⇒a2-ac+c2=b2Consider the RHS of 2cosA-C2=a+ca2-ac+c2
a+ca2-ac+c2=a+cb2=a+cb=sinA+sinCsinB               Using Sine Rule=2sinA+C2cosA-C2sinB=2sinBcosA-C2sinB           A+C2=B=2cosA-C2

IN A TRIANGLE ABC , THE ANGLE A, B, C ARE IN A.P. SHOW THAT 2 COS(A-C )/2 = (A+C)/ SQUARE ROOT OF (A SQUARE -AC +C SQUARE)