How does &#x03C0;<!-- π --> &#x2212;<!-- − --> arctan &#x2061;<!-- ⁡ --> (

Monfredo0n

Monfredo0n

Answered question

2022-05-26

How does π arctan ( 8 x ) arctan ( 13 20 x ) = arctan ( x 8 ) + arctan ( 20 x 13 )?

Answer & Explanation

Amelie Douglas

Amelie Douglas

Beginner2022-05-27Added 8 answers

Use the fact that for t 0, arctan ( t ) + arctan ( 1 / t ) = sgn ( t ) π 2
See here for the details: arctan ( x ) + arctan ( 1 / x ) = π 2
Then for x 0 and x 20,
arctan ( 8 x ) + arctan ( 13 20 x ) + arctan ( x 8 ) + arctan ( 20 x 13 ) = sgn ( x ) π 2 + sgn ( 20 x ) π 2 = { π if  0 < x < 20 , 0 if  x < 0  or  x > 20 .
istupilo8k

istupilo8k

Beginner2022-05-28Added 4 answers

There's a trigonometric identity for arctan: arctan ( α ) ± arctan ( β ) = arctan ( α ± β 1 α β )
In our case:
π arctan ( 8 x ) arctan ( 13 20 x ) = arctan ( x 8 ) + arctan ( 20 x 13 ) π arctan ( 13 20 x ) arctan ( 20 x 13 ) = arctan ( 8 x ) + arctan ( x 8 ) π ( arctan ( 13 20 x ) + arctan ( 20 x 13 ) ) = arctan ( 8 x ) + arctan ( x 8 ) π arctan ( 13 20 x + 20 x 13 1 ( 13 20 x 20 x 13 ) ) = arctan ( 8 x + x 8 1 ( 8 x x 8 ) ) π arctan ( 13 2 + ( 20 x ) 2 0 ) = arctan ( 8 2 + x 2 0 )
Note that the angle for l i m x arctan ( x ) = π 2 (in case x<0 we get π 2 )).
hence, we get: π π 2 = π 2

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