how do I do this? I think I have to simplify it?
(sin^2ө + cos^3ө/sinө + cosө)+ sinөcosө
To simplify the given expression, we can use trigonometric identities and algebraic manipulations. Let's break down each step:
Step 1: Simplify the numerator
Start by simplifying the numerator:
sin^2(θ) + cos^3(θ) = sin^2(θ) + cos^2(θ) * cos(θ)
= sin^2(θ) + (1 - sin^2(θ)) * cos(θ)
= sin^2(θ) + cos(θ) - sin^2(θ) * cos(θ)
= cos(θ) + sin^2(θ) * (1 - cos(θ))
Step 2: Simplify the denominator
The denominator is already in its simplified form.
Step 3: Combine the terms
Now, substitute the simplified numerator and denominator back into the expression:
(sin^2(θ) + cos^3(θ))/(sin(θ) + cos(θ)) + sin(θ) * cos(θ)
= (cos(θ) + sin^2(θ) * (1 - cos(θ)))/(sin(θ) + cos(θ)) + sin(θ) * cos(θ)
Step 4: Distribute and simplify
Distribute sin^2(θ) in the numerator:
(cos(θ) + sin^2(θ) - sin^2(θ) * cos(θ))/(sin(θ) + cos(θ)) + sin(θ) * cos(θ)
Step 5: Combine like terms
Combine the terms in the numerator:
(cos(θ) + sin^2(θ) - sin^2(θ) * cos(θ))/(sin(θ) + cos(θ)) + sin(θ) * cos(θ)
= (cos(θ) + sin^2(θ) - sin^2(θ) * cos(θ) + sin(θ) * cos(θ))/(sin(θ) + cos(θ))
At this point, we have simplified the given expression as much as possible using algebraic manipulations.