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

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

calculation:

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reason:

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2. use the trigonometric subtraction formula for sine to verify this identity:

sin((π / 2) – x) = cos x

calculation:

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reason:

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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. )

eve and marks ratios are equivalent

step-by-step explanation: