Prove this trigonometry equation: sin ⁡ 40 ∘ ⋅ sin ⁡ 50 ∘ is equal to 1 2 cos ⁡ 10 ∘

Question

Prove this trigonometry equation:   sin  ⁡      40    ∘    ⋅  sin  ⁡      50    ∘   is equal to       1    2    cos  ⁡      10    ∘

Ellie Meyers · Accepted Answer

Taking L.H.S.  s  i  n  40      °    s  i  n  50      °   or   s  i  n  50      °    s  i  n  40      °  We know the identity,   s  i  n  A  s  i  n  B  =            1      2                  (        c  o  s  (  A  −  B  )  −  c  o  s  (  A  +  B  )            )      Here we consider A=50° and B=40°Then,   s  i  n  50      °    s  i  n  40      °    =            1      2                  (        c  o  s  (  50      °    −  40      °    )  −  c  o  s  (  50      °    +  40      °    )            )        =            1      2                  (        c  o  s    10      °    −  c  o  s    90      °              )        =            1      2                  (        c  o  s    10      °    −  0            )                  (  ∵  c  o  s    90      °    =  0  )  =            1      2        c  o  s    10      °  L.H.S.=R.H.S.

measgachyx5q9 · Accepted Answer

Here,  L  .  H  .  S  =  s  i  n  40      °    .  s  i  n  50      °    =      2    2    .  s  i  n  40      °    .  s  i  n  50      °    =      1    2    (  c  o  s  10      °    −  c  o  s  90      °    )  =      1    2    c  o  s  10      °    =  R.  H  .  SProved.

Prove this trigonometry equation: sin ⁡ 40 ∘

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