Verified answer

Question :- Evaluate the following :-

[tex]\rm \: cos\bigg[ {tan}^{ - 1}\bigg(cot\bigg\lgroup \: {sin}^{ - 1} \dfrac{1}{2}\bigg\rgroup \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: \: \: \: \: (a) \: \: \: 1 \\ [/tex]

[tex]\rm \: \: \: \: \: (b) \: \: \: \frac{1}{4} \\ [/tex]

[tex]\rm \: \: \: \: \: (c) \: \: \: \frac{1}{8} \\ [/tex]

[tex]\rm \: \: \: \: \: (d) \: \: \: \frac{1}{2} \\ \\ \\ [/tex]

[tex]\large\underline{\sf{Solution-}}[/tex]

Given expression is

[tex]\rm \: cos\bigg[ {tan}^{ - 1}\bigg(cot\bigg\lgroup \: {sin}^{ - 1} \dfrac{1}{2}\bigg\rgroup \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: cos\bigg[ {tan}^{ - 1}\bigg(cot\bigg\lgroup \: {sin}^{ - 1} \bigg(sin\dfrac{\pi}{6}\bigg)\bigg\rgroup \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: = \: cos\bigg[ {tan}^{ - 1}\bigg(cot\bigg\lgroup \: \dfrac{\pi}{6}\bigg\rgroup \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: = \: cos\bigg[ {tan}^{ - 1}\bigg( \sqrt{3} \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: = \: cos\bigg[ {tan}^{ - 1}\bigg(tan\dfrac{\pi}{3} \bigg)\bigg] \\ \\ [/tex]

[tex]\rm \: = \: cos\bigg(\dfrac{\pi}{3}\bigg) \\ \\ [/tex]

[tex]\rm \: = \: \dfrac{1}{2} \\ \\ [/tex]

Hence,

[tex]\bf\implies \: cos\bigg[ {tan}^{ - 1}\bigg(cot\bigg\lgroup \: {sin}^{ - 1} \dfrac{1}{2}\bigg\rgroup \bigg)\bigg] = \dfrac{1}{2} \\ \\ [/tex]

So, option (d) is correct.

[tex]\rule{190pt}{2pt}[/tex]

[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Range \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf y =  {sin}^{ - 1}(sinx) & \sf  x \:  \: if -\dfrac{\pi  }{2} \leqslant x \leqslant \dfrac{\pi  }{2}\\ \\ \sf y =  {cos}^{ - 1}(cosx) & \sf x \:  \: if \: 0 \leqslant y \leqslant \pi \\ \\ \sf y =  {tan}^{ - 1}(tanx) & \sf x \:  \: if \:  - \dfrac{\pi  }{2} < x < \dfrac{\pi  }{2}\\ \\ \sf y =  {cosec}^{ - 1}(cosecx) & \sf x \:  \: if \: x \:  \in \: \bigg[ - \dfrac{\pi}{2}, \: \dfrac{\pi  }{2}\bigg] -  \{0 \}\\ \\ \sf y =  {sec}^{ - 1}(secx) & \sf x \:  \: if \: x \:  \in \: [0, \: \pi] \:   -  \: \bigg\{\dfrac{\pi  }{2}\bigg\}\\ \\ \sf y =  {cot}^{ - 1}(cotx) & \sf x \:  \: if \:  \:  \in \: \bigg( -  \dfrac{\pi  }{2} , \dfrac{\pi  }{2}\bigg) -  \{0 \} \end{array}} \\ \end{gathered} \\[/tex]

[tex]ok[/tex]

[tex]i \: am \: also \: from \: gujarat \: ahmedabad[/tex]