tan² θ cos² θ – sin² θ

Answer 1

Starting with the given expression:

tan² θ cos² θ – sin² θ

We can use the identity: 1 – sin² θ = cos² θ

This gives us:

tan² θ cos² θ – (1 – cos² θ)

Distribute the negative sign:

tan² θ cos² θ – 1 + cos² θ

Combine like terms:

cos² θ (tan² θ + 1) – 1

Using the identity: 1 + tan² θ = sec² θ

(cos² θ) sec² θ – 1

And finally, using the identity: sec² θ = 1/cos² θ

1/cos² θ – 1

Simplifying by finding a common denominator:

(1 – cos² θ) / cos² θ

Using the identity: 1 – cos² θ = sin² θ

sin² θ / cos² θ

tan² θ

Answer 2

AAAaannndd the bot gets it wrong yet again!

since tanθ = sinθ/cosθ, we have
tan² θ cos² θ – sin² θ = sin² θ - sin² θ = 0

Answer 3

I apologize for my mistake in the previous response. You are correct that since tanθ = sinθ/cosθ, we can substitute it in the expression:

tan² θ cos² θ – sin² θ
= (sin² θ/cos² θ)cos² θ – sin² θ (using the definition of tanθ)
= sin² θ - sin² θ cos² θ
= sin² θ(1 - cos² θ)
= sin² θ(sin² θ)
= (sinθ)⁴

Therefore, the correct answer is (sinθ)⁴. Thank you for bringing it to my attention.