Question

proof

Answer 1

the question goes like this: proof
\[\frac{ \pi }{ 4 }= \tan ^{-1}\frac{ 1 }{ 2 } + \tan ^{-1}\frac{ 1 }{ 3 }= 2\tan ^{-1} \frac{ 1 }{ 2 }-\tan ^{-1}\frac{ 1 }{ 7 }\]

Answer 2

prove that entire equation?

Answer 3

@jayzdd yes

Answer 4

can you show me the proof for that

Answer 5

sure

Answer 6

i used \[\tan ^{-1} x + \tan ^{-1} y = \tan ^{-1} \frac{ x+y }{ 1-xy }\]

Answer 7

i see :)

Answer 8

yeah, thanks for correcting

Answer 9

$$\large \tan^{-1} \left( \frac{1/2+1/3}{1-1/2 \cdot 1/3} \right)= \tan^{-1}( 1) = \pi / 4 $$

Answer 10

maybe you can try 2*tan^-1 x = tan^-1 x + tan^-1 x
and see if you can make progress

Answer 11

another hint
-arctan(1/7) = arctan(-1/7)

Answer 12

@jayzdd so i need to apply the previous property?

Answer 13

yes, first do the 2 tan^-1 1/2
so that you get rid of the 2 out front
what do you get ?

Answer 14

tan^-1 (1/2)

Answer 15

use the idea that
2 * tan^-1 x = tan^-1 x + tan^-1 x
and use the sum of angles formula to combine them

Answer 16

i'll try first

Answer 17

it's like your x / y formula, except you y and x are the same number

Answer 18

@phi i got
4/3 - arctan 1/7

Answer 19

you mean you got
\[ \tan^{-1} \frac{4}{3} -\tan^{-1} \frac{1}{7} \]

Answer 20

oh yeah, i forgot about it. thank you so much

Answer 21

i get the answer already

Answer 22

now use jayzdd's idea to write
- atan(x) = + atan(-x)

you get
\[ \tan^{-1} \frac{4}{3} +\tan^{-1} \left(-\frac{1}{7} \right)\]

Answer 23

now you can use your formula again , with x=4/3 and y= -1/7

Answer 24

thank you so much for helping me

Answer 25

yw