Final answer:
Statements b), c), and d) are true based on the Pythagorean identities, which include relationships between sine, cosine, tangent, secant, cosecant, and cotangent functions. Statement a) is false as it does not form a correct identity.
Step-by-step explanation:
The question asks which statement(s) are true based on the Pythagorean identities. We know that the fundamental Pythagorean identity is sin²(theta) + cos²(theta) = 1. From this identity, we can derive additional identities related to the tangent and secant functions. Option b) tan²(theta) + 1 = sec²(theta) is true because we can express tan(theta) as sin(theta)/cos(theta) and sec(theta) as 1/cos(theta), which upon squaring both gives us the said identity. Option c) 1 = sec²(theta) - tan²(theta) is also true because it is the rearrangement of the previous identity. Finally, option d) 1 + cot²(theta) = csc²(theta) is true as well, since cot(theta) is the reciprocal of tan(theta) and csc(theta) is the reciprocal of sin(theta), which also leads to a Pythagorean relationship. Option a) is false because it is not a simplification or a rearrangement of a Pythagorean identity. Therefore, statements b), c), and d) are true based on the Pythagorean identities.