tan² θ cos² θ – sin² θ
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
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
To simplify the expression tan² θ cos² θ – sin² θ, we can use trigonometric identities.
Step 1: Rewrite tan² θ as (sin θ / cos θ)².
(tan θ)² = (sin θ / cos θ)²
Step 2: Expand the expression (sin θ / cos θ)².
(tan θ)² = (sin θ)² / (cos θ)²
Step 3: Simplify the expression (sin θ)² / (cos θ)².
(tan θ)² = sin² θ / cos² θ
Step 4: Substitute (tan θ)² with sin² θ / cos² θ in the original expression.
sin² θ / cos² θ - sin² θ
Step 5: Combine like terms.
(sin² θ - sin² θ) / cos² θ
Step 6: Simplify the numerator.
0 / cos² θ
Step 7: The simplified expression is 0.
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.