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Prove that: tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4 - Mathematics

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Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`

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Solution

To prove `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4` we will use the following formula 

`tan^(-1)+tan^(-1)y=tan^(-1)((x+y)/(1-xy)),xy<1`

`Let S=tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)`

`S=[tan^(-1)(1/2)+tan^(-1)(1/5)]+tan^(-1)(1/8)`

`S=tan^(-1)((1/2+1/5)/(1-1/2 xx 1/5))+tan^(-1)(1/8)`

`S=tan^(-1)(7/9)+tan^(-1)(1/8)`

`=tan^(-1)((7/9+1/8)/(1-(7/9)xx(1/8)))`

`=tan^(-1)((56+9)/(72-7))`

`S=tan^(-1)(65/65)=tan^(-1)1=pi/4`

Hence, `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`

Concept: Properties of Inverse Trigonometric Functions
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