Question

prove that tan-1 +tan-2 +tan-3 =pi

Answer 1

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Answer 2

Given:

$tan^{-1}(1)+tan^{-1}(2)+tan^{-1}(3)$

To prove that

$tan^{-1}(1)+tan^{-1}(2)+tan^{-1}(3)=\pi$

Proof:

LHS

$tan^{-1}(1)+tan^{-1}(2)+tan^{-1}(3)$

$tan^{-1}(\frac{1+2}{1-1\times 2})+tan^{-1}(3)$

Using the formula

$tan^{-1}x+tan^{-1}y=tan^{-1}\frac{x+y}{1-xy}$

$tan^{-1}(-3)+tan^{-1}(3)$

Again using the formula

$tan^{-1}\frac{-3+3}{1+9}$

$tan^{-1}(0)$

$tan^{-1}tan(\pi)=\pi$

LHS=RHS

Hence, proved.