Question

Hi can anyone help with this question? I have been stuck on this for hours

Answer 1

• ab = - 1

Explanation:

Given

Equations:

• 1 - a tan θ = sec θ,
• b - cot θ = cosec θ.

To find

• The value of ab

Solution

Recall the identities:

• tan θ = sin θ / cos θ
• sec θ = 1 / cos θ
• cot θ = cos θ/ sin θ
• cosec θ = 1 / sin θ
• sin² θ + cos² θ = 1
• (a + b)(a - b) = a² - b²

Now find a and b, then find their product.

• 1 - a tan θ = sec θ
• 1 - a sin θ / cos θ = 1 / cos θ
• cos θ - a sin θ = 1
• a sin θ = cos θ - 1
• a = (cos θ - 1) / sin θ

• b - cot θ = cosec θ
• b - cos θ/ sin θ = 1 / sin θ
• b = cos θ/ sin θ + 1 / sin θ
• b = (cos θ + 1) / sin θ

• ab =
• (cos θ - 1) / sin θ × (cos θ + 1) / sin θ =
• (cos² θ - 1) / sin² θ =
• - sin² θ / sin² θ =
• - 1