Given expression: (sec θ - cosec θ) (1 tan θ cot θ)

To prove: sec θ. tan θ - cosec θ - cot θ

Proof:

Let's simplify the given expression step by step:

Step 1: Expand the expression (sec θ - cosec θ) (1 tan θ cot θ)

Using the distributive property of multiplication, we can expand the expression as follows:

(sec θ - cosec θ) (1 tan θ cot θ) = sec θ (1 tan θ cot θ) - cosec θ (1 tan θ cot θ)

Step 2: Simplify the expression sec θ (1 tan θ cot θ)

Using the trigonometric identity tan θ = sin θ / cos θ and cot θ = cos θ / sin θ, we can simplify the expression as follows:

sec θ (1 tan θ cot θ) = sec θ (1 * sin θ / cos θ * cos θ / sin θ)

Canceling out the common factors, we get:

sec θ (1 * sin θ / cos θ * cos θ / sin θ) = sec θ (sin θ / sin θ) = sec θ

Step 3: Simplify the expression cosec θ (1 tan θ cot θ)

Using the trigonometric identity tan θ = sin θ / cos θ and cot θ = cos θ / sin θ, we can simplify the expression as follows:

cosec θ (1 tan θ cot θ) = cosec θ (1 * sin θ / cos θ * cos θ / sin θ)

Canceling out the common factors, we get:

cosec θ (1 * sin θ / cos θ * cos θ / sin θ) = cosec θ (cos θ / cos θ) = cosec θ

Step 4: Simplify the expression -cosec θ - cot θ

Using the trigonometric identity cot θ = cos θ / sin θ, we can simplify the expression as follows:

-cosec θ - cot θ = -cosec θ - cos θ / sin θ

To combine the terms, we need a common denominator. Multiply the second term by sin θ / sin θ:

-cosec θ - cos θ / sin θ = -cosec θ * sin θ / sin θ - cos θ * sin θ / (sin θ * sin θ)

Simplifying further:

-cosec θ * sin θ / sin θ - cos θ * sin θ / (sin θ * sin θ) = -sin θ - cos θ / sin θ

Step 5: Combine the simplified expressions

Now, we can combine the simplified expressions obtained from Steps 2, 3, and 4:

sec θ - cosec θ - cot θ = sec