Prove: csc ⁡ a + cot ⁡ a = cot ⁡ a 2 All I have right now, from trig identities, is 1 sin ⁡ a + 1 tan ⁡ a = 1 tan ⁡ ( a / 2 ) Where do I go from there?

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Prove:   csc  ⁡  a  +  cot  ⁡  a  =  cot  ⁡      a    2  All I have right now, from trig identities, is      1          sin      ⁡      a        +      1          tan      ⁡      a        =      1          tan      ⁡      (      a              /            2      )      Where do I go from there?

glapaso7ng5 · Accepted Answer

We start with the following identities:     sin  ⁡  (  2  a  )  =  2  sin  ⁡  a  cos  ⁡  a       cos  ⁡  (  2  a  )  =  1  −  2      sin    2    ⁡  a  We solve these to get the half-angle identities:     sin  ⁡  (  a  )  =  2  sin  ⁡      a    2    cos  ⁡      a    2                 sin    2    ⁡      a    2    =      1    2    (  1  −  cos  ⁡  a  )We now tackle the problem      1          sin      ⁡      a        +            cos      ⁡      a              sin      ⁡      a        =            cos      ⁡              a        2                    sin      ⁡              a        2            multiplying out both sides we get that  2      sin    2    ⁡      a    2    (  cos  ⁡  a  +  1  )  =  2  sin  ⁡      a    2    cos  ⁡      a    2    sin  ⁡  aUsing the identities above we get that  (  1  −  cos  ⁡  a  )  (  1  +  cos  ⁡  a  )  =      sin    2    ⁡  a    ⟹    1  −      cos    2    ⁡  a  =      sin    2    ⁡  a    ⟹    1  =      sin    2    ⁡  a  +      cos    2    ⁡  aWe now have a trivial trigonometric identity, so the equivalence is proved

poklanima5lqp3 · Accepted Answer

csc  ⁡  a  +  cot  ⁡  a  =  cot  ⁡      a    2  L.H.S.    ⟹              1              sin        ⁡        a              +                  cos        ⁡        a                    sin        ⁡        a                  ⟹                      1        +        cos        ⁡        a                    sin        ⁡        a                  ⟹                      1        +        2                  cos          2                ⁡                  a          2                −        1                    sin        ⁡        a                  ⟹                      2                  cos          2                ⁡                  a          2                            2        sin        ⁡                  a          2                cos        ⁡                  a          2                          ⟹    cot  ⁡      a    2    =  R.H.S

Prove: csc ⁡ a + cot ⁡ a = cot ⁡<!-- ⁡

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