Prove that cot ⁡ 99 ∘ = 3 cot ⁡ 36 ∘ − sec ⁡ 18 ∘ − csc ⁡ 18 ∘

Question

Prove that   cot  ⁡      99    ∘    =  3  cot  ⁡      36    ∘    −  sec  ⁡      18    ∘    −  csc  ⁡      18    ∘

Kalmukujobvg · Accepted Answer

As   cot  ⁡  (      90    ∘    +  y  )  =  −  tan  ⁡  y  ,            1              sin        ⁡        2        x              −  tan  ⁡  x  =                    1        −        2                  sin          2                ⁡        x                    sin        ⁡        2        x              =  cot  ⁡  2  xWe need  cot  ⁡  2  x  +            1              cos        ⁡        2        x              =                    3        cos        ⁡        4        x                    sin        ⁡        4        x                ⟺    2      cos    2    ⁡  2  x  +  2  sin  ⁡  2  x  =  3  cos  ⁡  4  xIf   s  =  sin  ⁡  2  x  ,    ⟺    2  (  1  −      s    2    )  +  2  s  =  3  (  1  −  2      s    2    )  4      s    2    +  2  s  −  1  =  0So,   s  =  sin  ⁡      18    ∘   or   =  −  sin  ⁡      54    ∘

Prove that cot 99^(@)=3 cot 36^(@)−sec 18^(@)−csc 18^(@)

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