Question

1. Derive this identity from the sum and difference formulas for cosine:

Answer 1

Answer:

See below.

Step-by-step explanation:

1.  (1 / 2)[cos(a – b) – cos(a + b)]

= 1/2 ( cosa cosb + sina sinb - (cosa cosb - sina sinb)

= 1/2 ( cosa cosb - cosa cos b + sina sinb + sina sinb)

= 1/2 ( 2 sina sinb)

= sina sinb.

(I  used the 2 identities   cos(a - b) = cosa cosb + sina sinb) and

cos (a + b) = cosa cosb - sina sinb.)

2.  sin (π/2 - x)  = sin (π/2) cos x - cos(π/2) sin x

     =    1 * cos x - 0 * sinx  

     =   cosx - 0

     = cos x.

  (I used the identity sin(a - b) = sina cosb - cosa sinb

 and the fact that  sin(π/2) = 1 and cos (π/2) = 0. )

Answer 2

Answer:

1. sin a sin b = (1 / 2)[cos(a – b) – cos(a + b)]

Calculation:

Taking L.H.S. of above equation

                    (1 / 2)[cos(a – b) – cos(a + b)]

                 = (1 / 2) [ (cos a cos b + sin a sin b) - (cos a cos b - sin a sin b)]

                           {∵ cos(a – b) = cos a cos b + sin a sin b     &    cos(a + b) = cos a cos b - sin a sin b}

                 = (1 / 2) [ cos a cos b + sin a sin b - cos a cos b + sin a sin b]

                 = (1 / 2) [2 sin a sin b]

                 = sin a sin b

2. sin((π / 2) – x) = cos x

Calculation:

sin((π / 2) – x) = sin (π / 2) cos x - cos (π / 2) sin x

                                                      {∵sin(a - b) = sin a cos b - cos b sin a                                                    

                                                       &  sin (π / 2)  =  1   &  cos (π / 2) = 0}

                       = 1 × cos x - 0 × sin x

                       = cos x