Final answer:
To simplify the given expression, we expand and combine like terms. The expression (cosecθ + cosθ)^2 + (secθ + sinθ)^2 simplifies to 4 + 2sinθ/cosθ + 2cosθ/sinθ.
Step-by-step explanation:
To simplify the given expression, we expand and combine like terms:
Using the identity (a + b)^2 = a^2 + 2ab + b^2,
(cosecθ + cosθ)^2 + (secθ + sinθ)^2 = cosec^2θ + 2cosecθcosθ + cos^2θ + sec^2θ + 2secθsinθ + sin^2θ
Using the trigonometric identities cosecθ = 1/sinθ and secθ = 1/cosθ,
cosec^2θ + 2cosecθcosθ + cos^2θ + sec^2θ + 2secθsinθ + sin^2θ = (1/sin^2θ) + 2(1/sinθ)(cosθ) + cos^2θ + (1/cos^2θ) + 2(1/cosθ)(sinθ) + sin^2θ
Simplifying further,
(1/sin^2θ) + 2(1/sinθ)(cosθ) + cos^2θ + (1/cos^2θ) + 2(1/cosθ)(sinθ) + sin^2θ = 1/sin^2θ + 2cosθ/sinθ + cos^2θ + 1/cos^2θ + 2sinθ/cosθ + sin^2θ
Using the Pythagorean identities sin^2θ + cos^2θ = 1, we can simplify the expression further:
1/sin^2θ + 2cosθ/sinθ + cos^2θ + 1/cos^2θ + 2sinθ/cosθ + sin^2θ = 1/sin^2θ + 2cosθ/sinθ + 1 + 1/cos^2θ + 2sinθ/cosθ + 1
Combining like terms, we have:
(cosecθ + cosθ)^2 + (secθ + sinθ)^2 = 4 + 2sinθ/cosθ + 2cosθ/sinθ