Question

There exists a positive real number x satisfying cos(tan1x) = x. The value of cos1(x 22) is

Answer 1

Answer:

Step-by-step explanation:

Given condition:cos(tan inverse x)=x

So, cos inverse[cos(tan inverse x)]=cos inverse x.

So, tan inverse x=cos inverse x.{cos inverse(cos x)=x}

Or, tan inverse x+sin inverse x=π/2.{cos inverse x+sin inverse x=π/2}

Therefore, tan inverse x+tan inverse[x/√(1-x^2)]=π/2.{sin inverse x=tan inverse[x/√(1-x^2)]

Solving this equation, we get x^2=(√5–1)/2.

So, cos inverse(x^2/2)=cos inverse[(√5–1)/4]=72°