Finding solution of trigonometric equationHow to find the solutions of this trigonometric equation ∑ m = 1 6 c o s e c ( θ + ( m − 1 ) π 4 ) c o s e c ( θ + m π 4 ) = 4 2 if 0 &lt; θ &lt; π / 2

Question

Finding solution of trigonometric equationHow to find the solutions of this trigonometric equation      ∑          m      =      1        6    c  o  s  e  c  (  θ  +            (      m      −      1      )      π        4    )  c  o  s  e  c  (  θ  +            m      π        4    )  =  4      2  if   0  &amp;lt;  θ  &amp;lt;  π      /    2

Kaylee Evans · Accepted Answer

HINT:  csc  ⁡      (    θ    +                            (          m          −          1          )          π                4              )    ⋅  csc  ⁡      (    θ    +                            m          π                4              )    =            1              sin        ⁡                  (          θ          +                                                    (                m                −                1                )                π                            4                                )                ⋅        sin        ⁡                  (          θ          +                                                    m                π                            4                                )                                        sin        ⁡        (        A        −        B        )                    sin        ⁡        A        sin        ⁡        B              =                    sin        ⁡        A        cos        ⁡        B        −        cos        ⁡        A        sin        ⁡        B                    sin        ⁡        A        sin        ⁡        B              =  cot  ⁡  B  −  cot  ⁡  AHere   A  =  θ  +                    m        π            4       and   B  =  θ  +                    (        m        −        1        )        π            4        ,  A  −  B  =            π      4          ⟹              1              sin        ⁡                  (          θ          +                                                    (                m                −                1                )                π                            4                                )                ⋅        sin        ⁡                  (          θ          +                                                    m                π                            4                                )                      =            1              sin        ⁡                              π            4                                ⋅                    sin        ⁡                  {          θ          +                                                    m                π                            4                                −                      (            θ            +                                                            (                  m                  −                  1                  )                  π                                4                                      )                    }                            sin        ⁡                  (          θ          +                                                    (                m                −                1                )                π                            4                                )                ⋅        sin        ⁡                  (          θ          +                                                    m                π                            4                                )                      =                    cot        ⁡                  (          θ          +                                                    (                m                −                1                )                π                            4                                )                −        cot        ⁡                  (          θ          +                                                    m                π                            4                                )                            sin        ⁡                              π            4                              Set   m  =  1  ,  2  ,  3  ,  4  ,  5  ,  6 and add and finally simplify.

Finding solution of trigonometric equation sum_(m=1)^6 cosec(theta + ((m-1) pi)/(4)) cosec(theta + (m pi)/(4)) =4 sqrt(2) if 0<theta<pi/2.

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