How to find the solution of this trigonometric equation cot ⁡ ( x + 110 ∘ ) = cot ⁡ ( x + 60 ∘ ) cot ⁡ x cot ⁡ ( x − 60 ∘ )I have used the formulae cos ⁡ ( x + 60 ∘ ) cos ⁡ ( x − 60 ∘ ) = cos 2 ⁡ 60 ∘ − sin 2 ⁡ x sin ⁡ ( x + 60 ∘ ) sin ⁡ ( x − 60 ∘ ) = sin 2 ⁡ x − sin 2 ⁡ 60 ∘ How to move further? What is the least positive value of x?

Question

How to find the solution of this trigonometric equation  cot  ⁡  (  x  +      110    ∘    )  =  cot  ⁡  (  x  +      60    ∘    )  cot  ⁡  x  cot  ⁡  (  x  −      60    ∘    )I have used the formulae  cos  ⁡  (  x  +      60    ∘    )  cos  ⁡  (  x  −      60    ∘    )  =      cos    2    ⁡      60    ∘    −      sin    2    ⁡  x  sin  ⁡  (  x  +      60    ∘    )  sin  ⁡  (  x  −      60    ∘    )  =      sin    2    ⁡  x  −      sin    2    ⁡      60    ∘  How to move further? What is the least positive value of x?

Brendan Bush · Accepted Answer

cot  ⁡  (  x  +      110    ∘    )  =  cot  ⁡  (  x  +      60    ∘    )  cot  ⁡  x  cot  ⁡  (  x  −      60    ∘    )            cot      ⁡      (      x      +              110        ∘            )              cot      ⁡      x        =  cot  ⁡  (  x  +      60    ∘    )  ⋅  cot  ⁡  (  x  −      60    ∘    )            cos      ⁡      (      x      +              110        ∘            )      ⋅      sin      ⁡      x              sin      ⁡      (      x      +              110        ∘            )      ⋅      cos      ⁡      x        =            cos      ⁡      (      x      +              60        ∘            )      ⋅      cos      ⁡      (      x      −              60        ∘            )              sin      ⁡      (      x      +              60        ∘            )      ⋅      sin      ⁡      (      x      −              60        ∘            )      Now Using Componendo and Dividendo, We get            cos      ⁡      (      x      +              110        ∘            )      ⋅      sin      ⁡      x      +      sin      ⁡      (      x      +              110        ∘            )      ⋅      cos      ⁡      x              cos      ⁡      (      x      +              110        ∘            )      ⋅      sin      ⁡      x      −      sin      ⁡      (      x      +              110        ∘            )      ⋅      cos      ⁡      x        =            cos      ⁡      (      x      +              60        ∘            )      ⋅      cos      ⁡      (      x      −              60        ∘            )      +      sin      ⁡      (      x      +              60        ∘            )      ⋅      sin      ⁡      (      x      −              60        ∘            )              cos      ⁡      (      x      +              60        ∘            )      ⋅      cos      ⁡      (      x      −              60        ∘            )      −      sin      ⁡      (      x      +              60        ∘            )      ⋅      sin      ⁡      (      x      −              60        ∘            )        −            sin      ⁡              [        (        x        +                  110          ∘                )        +        x        ]                    sin      ⁡              [        (        x        +                  110          ∘                )        −        x        ]              =            cos      ⁡      [      (      x      +              60        ∘            )      −      (      x      −              60        ∘            )      ]              cos      ⁡      [      (      x      +              60        ∘            )      +      (      x      −              60        ∘            )      ]      So we get  −            sin      ⁡      (      2      x      +              110        ∘            )              sin      ⁡      (              110        ∘            )        =            cos      ⁡      (              120        ∘            )              cos      ⁡      (      2      x      )        sin  ⁡  (  2  x  +      110    ∘    )  ⋅  cos  ⁡  (  2  x  )  =      1    2    sin  ⁡  (      110    0    )So we get  2  sin  ⁡  (  2  x  +      110    ∘    )  ⋅  cos  ⁡  (  2  x  )  =  sin  ⁡  (      110    0    )  sin  ⁡  (  4  x  +      110    ∘    )  +  sin  ⁡  (      110    ∘    )  =  sin  ⁡  (      110    ∘    )So we get  sin  ⁡  (  4  x  +      110    ∘    )  =  0  ⇒  4  x  +      110    0    =  n  ×      180    ∘      ,Where   n  ∈      Z

How to find the solution of this trigonometric equation cot ⁡ ( x

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