Question

value of tan^-1(√3) - sec^-1(-2)​

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: {tan}^{ - 1} \sqrt{3} - {sec}^{ -1 } ( - 2)$

Identities Used :-

$\boxed{ \red{ \bf \: tan\dfrac{\pi}{ 3} = \sqrt{3}}}$

$\boxed{ \red{ \bf \: sec\dfrac{\pi}{ 3} = 2}}$

$\boxed{ \red{ \bf \: {tan}^{ - 1}(tanx) = x}}$

$\boxed{ \red{ \bf \: {sec}^{ - 1}(secx) = x}}$

$\boxed{ \red{ \bf \: {sec}^{ - 1} ( - x) = \pi \: - {sec}^{ - 1} x}}$

$\huge\underline{\sf{Solution-}}$

$\rm :\longmapsto\: {tan}^{ - 1} \sqrt{3} - {sec}^{ -1 } ( - 2)$

$\rm :\longmapsto\: = \: \: {tan}^{ - 1} \sqrt{3} - \bigg(\pi \: - {sec}^{ -1 } 2 \bigg)$

$\rm :\longmapsto\: = \: \: {tan}^{ - 1} \bigg(tan\dfrac{\pi}{ 3} \bigg) - \pi \: + {sec}^{ - 1}\bigg(sec\dfrac{\pi}{ 3} \bigg)$

$\rm :\longmapsto\: = \: \: \dfrac{\pi}{ 3} - \pi \: + \dfrac{\pi}{ 3}$

$\rm :\longmapsto\: = \: \: \dfrac{2\pi}{ 3} - \pi \:$

$\rm :\longmapsto\: = \: \: \dfrac{2\pi \: - 3\pi}{ 3}$

$\rm :\longmapsto\: = \: \: \dfrac{\: - \pi}{ 3}$

Additional Information :-

$\boxed{ \red{ \bf \: {cos}^{ - 1} ( - x) = \pi \: - {cos}^{ - 1} x}}$

$\boxed{ \red{ \bf \: {cot}^{ - 1} ( - x) = \pi \: - {cot}^{ - 1} x}}$

$\boxed{ \red{ \bf \: {sin}^{ - 1}( - x) = - {sin}^{ - 1}x}}$

$\boxed{ \red{ \bf \: {tan}^{ - 1}( - x) = - {tan}^{ - 1}x}}$

$\boxed{ \red{ \bf \: {cosec}^{ - 1}( - x) = - {cosec}^{ - 1}x}}$