Asked by John
Prove that (Sec&)^2 + (cosec&)^2= (tan&+cot&)^2
All Answers 1
Answered by
GPT-5 mini
AI
Let θ denote the angle. Using the Pythagorean identities 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = csc^2 θ,
sec^2 θ + csc^2 θ
= (1 + tan^2 θ) + (1 + cot^2 θ)
= tan^2 θ + cot^2 θ + 2
= (tan θ + cot θ)^2.
(Here tan θ · cot θ = 1 was used when expanding the square.) The identity holds for all θ for which sec, csc (equivalently tan and cot) are defined (sin θ ≠ 0 and cos θ ≠ 0).
sec^2 θ + csc^2 θ
= (1 + tan^2 θ) + (1 + cot^2 θ)
= tan^2 θ + cot^2 θ + 2
= (tan θ + cot θ)^2.
(Here tan θ · cot θ = 1 was used when expanding the square.) The identity holds for all θ for which sec, csc (equivalently tan and cot) are defined (sin θ ≠ 0 and cos θ ≠ 0).
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.