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Trigonometric Functions of Multiple and Submultiple Angles

 Trigonometric Functions

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Example 2: Prove that cos x1 + sin x = tan (4-x2) .

Solution: LHS = cos x1 + sin x = sin (2 - x)1 + cos (2 - x)

= 2 sin (4 - x2) cos (4 - x2)2 cos2 (4 - x2)

= tan (4-x2)  = RHS


Example 3: Show that 2+2+2+2 cos 8  = 2 cos

Solution: LHS = 2+2+2+2 cos 8

                        =2+2+2+2 (1+ cos 8)

   =2+2+(2cos 4)              (∵ cos 2ЁЭЬГ = 2 cos2 -1)

                 =2+4cos22                         (∵ cos 2ЁЭЬГ = 2 cos2 -1)

                  =2+2cos2

                  = 2 cos

                  = RHS










Example 4: Prove that (cos A + cos B)^2 + (sin A- sin B)^2= 4 cos^2(A + B/2)

Solution: We have, LHS =(cos A + cos B)^2 + (sin A - sin B)^2 

        = cos^2A +cos^2B + 2 cos A cos B + (sin^2A + sin^2B - 2 sin A sin B)

        = cos^2A +sin^2A + cos^2B +sin^2B + 2 (cos A cosB -sin A sin B)                 

      = 2+2 cos (A+B)

        = 2[1+cos (A+B)]

        = 4 cos^2(A+B/2)


Summary

In this module some formulae of trigonometric functions of multiple and submultiple angles have been derived. Using these identities, some results have been proved.