Can someone tell me how to do this problem?
(sin^2 θ - cotθ * tan θ)/ cotθ * sin θ
Please and thank you.
a quick observation ...
tanØ x cotØ = 1
so (sin^2 θ - cotθ * tan θ)/ cotθ * sin θ
= (sin^2Ø - 1)(tanØ)(sinØ)
= (sin^2Ø - 1)(sinØ/cosØ)sinØ
= sin^2Ø - (sin^2Ø + cos^2Ø)(sin^2Ø/cosØ
= -cos^2Øsin^2Ø/cosØ
= -cosØsin^2Ø
Where did this part come from?
- (sin^2Ø + cos^2Ø)
To simplify the given expression (sin^2 θ - cotθ * tan θ) / cotθ * sin θ, we need to use some trigonometric identities and simplify each term separately.
Let's start with the numerator:
sin^2 θ - cotθ * tan θ
We know that cotθ = 1/tanθ, so we can substitute that in:
sin^2 θ - (1/tanθ) * tan θ
Also, recall the identity: sin^2 θ + cos^2 θ = 1
So, we can rewrite sin^2 θ as 1 - cos^2 θ:
(1 - cos^2 θ) - (1/tanθ) * tan θ
Now, we can simplify this expression further:
1 - cos^2 θ - 1
- cos^2 θ
Now, let's move on to the denominator:
cotθ * sin θ
We know that cotθ = 1/tanθ, so we can rewrite it as:
(1/tanθ) * sin θ
Now, we can simplify this expression further:
sin θ / tanθ
Finally, we can put the simplified numerator and denominator back together:
- cos^2 θ / (sin θ / tanθ)
To simplify further, we can multiply the numerator by the reciprocal of the denominator:
- cos^2 θ * (tanθ / sin θ)
Now, we have simplified the expression to -cos^2 θ * tanθ / sin θ.
So, the simplified form of the given expression is -cos^2 θ * tanθ / sin θ.