To answer the question, let's consider the known Pythagorean identities in trigonometry. These identities relate the square of the trigonometric functions to 1 and their respective reciprocals.
1. The first Pythagorean identity is:
[tex]\[
\sin^2(t) + \cos^2(t) = 1
\][/tex]
This identity is derived from the fundamental relationship in trigonometry where the sum of the squares of sine and cosine of an angle always equals 1.
2. The second Pythagorean identity involves the tangent function. It states:
[tex]\[
1 + \tan^2(t) = \sec^2(t)
\][/tex]
This is because the tangent function is the ratio of sine and cosine, and the secant function is the reciprocal of the cosine. Squaring these relationships and adding 1 gives the identity involving secant.
3. The third Pythagorean identity involves the cotangent function. It states:
[tex]\[
1 + \cot^2(t) = \csc^2(t)
\][/tex]
Here, the cotangent function is the ratio of cosine over sine, and the cosecant function is the reciprocal of the sine. Squaring these relationships and adding 1 results in the identity involving cosecant.
Therefore, the completed statements, based on the Pythagorean identities, are:
[tex]\(\sin^2(t) + \cos^2(t) = 1\)[/tex],
[tex]\(1 + \tan^2(t) = \sec^2(t)\)[/tex],
and [tex]\(1 + \cot^2(t) = \csc^2(t)\)[/tex].
