We want to determine which of the expressions can be used to form an identity with the given expression.
tan^2θ+1/tan^2θ
To find this expression, we will use Trigonometric Identities to simplify the given expression. First, we will use one of the Pythagorean Identities.
tan^2θ+1=sec^2θLet's use this identity to simplify the numerator of our expression.
tan^2θ+1= sec^2θ
⇓
tan^2θ+1/tan^2θ=sec^2θ/tan^2θ
In the next step we will use one of the Reciprocal Identities to transform the numerator a bit more.
secθ= 1/cosθ, for cosθ ≠0
⇓
sec^2θ/tan^2θ=( 1cosθ)^2/tan^2θ,for cosθ≠0
Now, to transform the denominator we will use one of the Quotient Identities.
tanθ= sinθ/cosθ, for cosθ ≠0
⇓
( 1cosθ)^2/tan^2θ=( 1cosθ)^2/( sinθcosθ)^2,for cosθ≠0
Let's now simplify the obtained expression. Note that it is true when cosθ does not equal 0.
Finally, recall another Reciprocal Identity considering the reciprocal of sine and transform it a bit to match the obtained expression.
1/sinθ=cscθ for sinθ ≠0
⇕
1/sin^2θ=csc^2θ for sinθ ≠0
Since all the transformations were based on equations, the obtained expression forms an identity with the given one. Note that during the process we stated that this is true for sinθ≠0 and cosθ≠0.
tan^2θ+1/tan^2θ=csc^2θ
This corresponds to option D.
