Subjective Type

Prove: $$\displaystyle { \tan }^{ -1 }\frac { 2 }{ 11 } +{ \tan }^{ -1 }\frac { 7 }{ 24 } ={ \tan }^{ -1 }\frac { 1 }{ 2 } $$

Solution

$$\text{L.H.S.}\displaystyle ={ \tan }^{ -1 }\frac { 2 }{ 11 } +{ \tan }^{ -1 }\frac { 7 }{ 24 } $$

$$\displaystyle ={ \tan }^{ -1 }\dfrac { \dfrac { 2 }{ 11 } +\dfrac { 7 }{ 24 } }{ 1-\dfrac
{ 2 }{ 11 } .\dfrac { 7 }{ 24 } }$$, .....$$\left[\because { \tan }^{ -1 }x+{ \tan }^{ -1 }y={ \tan }^{ -1 }\dfrac { x+y }{ 1-xy } \right] $$

$$\displaystyle ={ \tan }^{ -1 }\dfrac { \dfrac { 48+77 }{ 11\times 24 } }{ \dfrac { 11\times 24-14 }{ 11\times 24 } } $$

$$\displaystyle ={ \tan }^{ -1 }\frac { 48+77 }{ 264-14 } ={ \tan }^{ -1 }\frac { 125 }{
250 } ={ \tan }^{ -1 }\frac { 1 }{ 2 } $$

$$=\text{R.H.S.}$$

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