f  secθ=114sec𝜃=114, then  ∣∣tanθ2∣∣tan𝜃2 is equal to

Given that secθ = 114, we know that secθ is the reciprocal of cosθ. So, cosθ = 1/secθ = 1/114.

We also know that tanθ = sinθ/cosθ. But, sinθ can be found using the Pythagorean identity: sin²θ + cos²θ = 1. So, sinθ = sqrt(1 - cos²θ) = sqrt(1 - (1/114)²).

Now, we need to find tan(θ/2). We can use the half-angle formula for tangent, which is:

tan(θ/2) = ± sqrt((1 - cosθ) / (1 + cosθ)).

Substituting the value of cosθ we found earlier, we get:

tan(θ/2) = ± sqrt((1 - 1/114) / (1 + 1/114)).

Solving this will give us the value of tan(θ/2). The absolute value of this result will be the answer to the problem.