Solving inverse trig question without forming casesQuestion:Find x if arctan ⁡ ( x + 3 ) − arctan ⁡ ( x − 3 ) = arctan ⁡ ( 3 / 4 )My attempt:I know the formula: arctan ⁡ x − arctan ⁡ y = arctan ⁡ ( x − y 1 + x y )for both x , y &gt; 0If both x , y are NOT greater than 0 then we'll need to form a second case in the above formula.I solved assuming both x + 3 and x − 3 are greater than zero and got a quadratic in the end solving which I got x = ± 4I dutifully rejected x = − 4 as it invalidated my assumption. However on putting in the above equation I found that it DOES satisfy the equation.I do not wish to form cases for positive/negative. I wish to know if there is a simpler and direct way to solve such a question. Thanks!

Question

Solving inverse trig question without forming casesQuestion:Find   x if   arctan  ⁡  (  x  +  3  )  −  arctan  ⁡  (  x  −  3  )  =  arctan  ⁡  (  3      /    4  )My attempt:I know the formula:  arctan  ⁡  x  −  arctan  ⁡  y  =  arctan  ⁡  (            x      −      y              1      +      x      y        )for both   x  ,  y  &amp;gt;  0If both   x  ,  y are NOT greater than   0 then we&#039;ll need to form a second case in the above formula.I solved assuming both   x  +  3 and   x  −  3 are greater than zero and got a quadratic in the end solving which I got   x  =  ±  4I dutifully rejected   x  =  −  4 as it invalidated my assumption. However on putting in the above equation I found that it DOES satisfy the equation.I do not wish to form cases for positive/negative. I wish to know if there is a simpler and direct way to solve such a question. Thanks!

Vartavk · Accepted Answer

Your answer   {  4  ,  −  4  } is right.A full solution:Since   tan  ⁡  (  α  −  β  )  =            tan      ⁡      α      −      tan      ⁡      β              1      +      tan      ⁡      α      tan      ⁡      β       and   tan  ⁡  arctan  ⁡      x    =  x for all real   x, we obtain  tan  ⁡      (    arctan    ⁡    (    x    +    3    )    −    arctan    ⁡    (    x    −    3    )    )    =  tan  ⁡  arctan  ⁡      3    4  or            x      +      3      −      (      x      −      3      )              1      +      (      x      +      3      )      (      x      −      3      )        =      3    4  or      x    2    =  16  ,which gives   x  =  4 or   x  =  −  4The checking of these numbers gets that indeed, they are roots and we are done!

Find x if arctan(x+3)−arctan(x−3)=arctan(3/4)

Answered question

Answer & Explanation

New Questions in Pre-Algebra