Solve the equation tan 4 ⁡ x + tan 4 ⁡ y + 2 cot 2 ⁡ x cot 2 ⁡ y = 3 + sin 2 ⁡ ( x + y )for the values of x and y.Partial Solution: One can easily guess that x = π / 4 and y = π / 4 satisfy the given equation. So one solution is x = n π ± π / 4 , n ∈ z and y = n π ± π / 4 , n ∈ z . But how do we prove that this is the only solution?

Question

Solve the equation      tan    4    ⁡  x  +      tan    4    ⁡  y  +  2      cot    2    ⁡  x      cot    2    ⁡  y  =  3  +      sin    2    ⁡  (  x  +  y  )for the values of x and y.Partial Solution: One can easily guess that   x  =  π      /    4 and   y  =  π      /    4 satisfy the given equation. So one solution is   x  =  n  π  ±  π      /    4  ,  n  ∈      z   and   y  =  n  π  ±  π      /    4  ,  n  ∈      z  . But how do we prove that this is the only solution?

Colin Moran · Accepted Answer

Hint:AM-GM inequality says                              tan          4                ⁡        x        +                  tan          4                ⁡        y        +                  cot          2                ⁡        x                  cot          2                ⁡        y        +                  cot          2                ⁡        x                  cot          2                ⁡        y            4        ≥  1

Solve the equation tan 4 </msup> ⁡ x + tan 4 </ms

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