(sinθ−secθ) 2 +(cosθ−cosecθ) 2

Sure, let's solve this step by step:

First, let's remember the Pythagorean identities:

1. sin²θ + cos²θ = 1
2. sec²θ - tan²θ = 1
3. cosec²θ - cot²θ = 1

Now, let's expand the given expression:

(sinθ−secθ)² +(cosθ−cosecθ)² = sin²θ - 2sinθsecθ + sec²θ + cos²θ - 2cosθcosecθ + cosec²θ

We can rearrange this as:

= (sin²θ + cos²θ) + (sec²θ + cosec²θ) - 2sinθsecθ - 2cosθcosecθ

Now, we can substitute the Pythagorean identities:

= 1 + (1 + tan²θ + cot²θ) - 2sinθsecθ - 2cosθcosecθ

This is the simplified form of the given expression.