sin^2 theta + 2 cos theta -1/ (sin^2 theta +3 cos theta-3= cos^2 theta+cos theta/
(-sin^2 theta)
Prove the identity.
Start from the left, it would be easy to change sin²(θ) into cos²(θ) using the identity:
sin²(θ)+cos²(θ)=1
Do not forget to put parentheses around the numerator and denominator. The expression that you posted does not evaluate to what you meant.
(sin²(θ)+2cos(θ)-1) / (sin²(θ)+3cos(θ)-3)
=(2cos(θ)-cos²(θ)) /
(3cos(θ)-cos²(θ)-2)
=cos(θ)(2-cos(θ)) / ((cos(θ)-2)(1-cos(θ)))
=cos(θ)/(cos(θ)-1)
multiply by (1+cos(θ)) both numerator and denominator:
= cos(θ)(1+cos(θ)) / ((cos(θ)-1)(1+cos(θ)))
=-cos(θ)(1+cos(θ)) / sin²(θ)
To prove the given identity:
sin^2 theta + 2 cos theta -1 = cos^2 theta + cos theta / sin^2 theta + 3 cos theta -3 = -sin^2 theta
we'll need to manipulate the left-hand side (LHS) of the equation and the right-hand side (RHS) separately, and then equate them. Let's start with LHS:
LHS = sin^2 theta + 2 cos theta -1
We can write sin^2 theta as 1 - cos^2 theta, so substituting that in:
LHS = (1 - cos^2 theta) + 2 cos theta - 1
Simplifying:
LHS = -cos^2 theta + 2 cos theta
Now let's work on the RHS:
RHS = cos^2 theta + cos theta / sin^2 theta + 3 cos theta -3 = -sin^2 theta
We can simplify the denominator separately by factoring it as a quadratic equation:
sin^2 theta + 3 cos theta - 3 = (sin theta - 1)(sin theta + 3)
Now we can rewrite the RHS:
RHS = cos^2 theta + cos theta / (sin theta - 1)(sin theta + 3) = -sin^2 theta
Expanding the denominator and multiplying through by (sin theta - 1)(sin theta + 3):
RHS = (cos^2 theta + cos theta)(sin theta - 1)(sin theta + 3) = -sin^2 theta
Now let's simplify the RHS further:
RHS = cos^2 theta sin theta - cos theta + cos^3 theta + cos^2 theta - sin theta cos^2 theta - sin theta + 3 cos^2 theta + 3 cos theta =
cos^3 theta + 4 cos^2 theta - cos theta - sin theta + sin theta cos^2 theta = -sin^2 theta
Combining like terms:
RHS = cos^3 theta + 4 cos^2 theta - cos theta = -sin^2 theta
Now that we have simplified both the LHS and RHS, let's compare them:
LHS = -cos^2 theta + 2 cos theta
RHS = cos^3 theta + 4 cos^2 theta - cos theta
As you can see, the LHS and RHS are not equal. Therefore, the given equation is not an identity and cannot be proven.