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Question: There exists a positive real number x satisfying cos (tan<sup>–1</sup> x) = x, the value of cos<sup...

There exists a positive real number x satisfying cos (tan–1 x)

= x, the value of cos–1 is-

A

π10\frac { \pi } { 10 }

B

π5\frac { \pi } { 5 }

C

2π5\frac { 2 \pi } { 5 }

D

4π5\frac { 4 \pi } { 5 }

Answer

2π5\frac { 2 \pi } { 5 }

Explanation

Solution

cos–1x = tan–1x = cos–1 11+x2\frac { 1 } { \sqrt { 1 + x ^ { 2 } } }

Ž x =−1+52\frac { - 1 + \sqrt { 5 } } { 2 }

Ž x22\frac { x ^ { 2 } } { 2 } = 5−14\frac { \sqrt { 5 } - 1 } { 4 } cos–1 = 2π5\frac { 2 \pi } { 5 }