Final answer:
To find the values of the given trigonometric expressions, we can use the information provided and the definitions of the trigonometric functions. Using the provided value of tanθ=7, we can find the exact values of sec²θ, cotθ, cot(π/2 −θ), and csc²θ.
Step-by-step explanation:
To find the values of the given trigonometric expressions, we can use the information provided and the definitions of the trigonometric functions.
(a) To find sec²θ, we can use the identity sec²θ = 1/cos²θ. Since tanθ = 7, we can use the Pythagorean identity tan²θ + 1 = sec²θ. Substituting 7 for tanθ, we get 7² + 1 = sec²θ. Solving this equation, we find sec²θ = 50.
(b) To find cotθ, we can use the identity cotθ = 1/tanθ. Since tanθ = 7, we can conclude that cotθ = 1/7.
(c) To find cot(π/2 −θ), we can use the identity cot(π/2 −θ) = tanθ. Since tanθ = 7, cot(π/2 −θ) = 7.
(d) To find csc²θ, we can use the identity csc²θ = 1/sin²θ. Since tanθ = 7, we can use the Pythagorean identity tan²θ + 1 = 1/sin²θ. Substituting 7 for tanθ, we get 7² + 1 = 1/sin²θ. Solving this equation, we find sin²θ = 1/50. Therefore, csc²θ = 50.