Concept:Rewrite all trigonometric ratios in terms of sinθ and cosθ. Then simplify using algebraic identities and the Pythagorean identity.
Explanation:First, write tanθ = sinθ/cosθ, secθ = 1/cosθ, cotθ = cosθ/sinθ, cosecθ = 1/sinθ.
The expression becomes: (1 + sinθ/cosθ + 1/cosθ) × (1 + cosθ/sinθ - 1/sinθ).
Combine terms inside each bracket over a common denominator:
First bracket: (cosθ + sinθ + 1)/cosθ.
Second bracket: (sinθ + cosθ - 1)/sinθ.
Multiply the two fractions: [(cosθ + sinθ + 1)(cosθ + sinθ - 1)] / (cosθ sinθ).
Notice the numerator is of the form (a+1)(a-1) where a = cosθ + sinθ. This equals a² - 1.
So numerator = (cosθ + sinθ)² - 1 = cos²θ + sin²θ + 2cosθ sinθ - 1.
Using cos²θ + sin²θ = 1, numerator simplifies to 1 + 2cosθ sinθ - 1 = 2cosθ sinθ.
Thus the whole expression = (2cosθ sinθ) / (cosθ sinθ) = 2.
Answer:2 (Option C)