14. a) Which equations are not identities? Justify your answers.
b) For those equations that are identities, state any restrictions on the variables.
i)
Left Side
Edit: Was an error made here?
Right Side:
Left Side = Right Side
Therefore, this equation is an identity.
When I showed this solution to my teacher, I was told that had I used identities that we haven't learned yet. She said that on the exam, we were to only be able to use reciprocal identities and the Pythagorean identity She also told me that using these identities, I can make the and equal however with different values. Is there another way of proving this equation to be an identity?
Also, how do I know whether or not an equation will eventually end up as ? The only reason I know it with the few assigned questions is because there is an answer at the back of the book. Though on an exam I won't have access to these answers. Is it just trial and error? Because if it is, I can be going on for a pretty long time just trying to solve one question.
When proving identities it is good practice to only change one side and manipulate it to get the other.
I have put my steps as spoilers along with explanation.
From the LHS
Use FOIL
Spoiler:
Multiply through by as this is the LCD of the expression.
Spoiler:
Using the identity rewrite any cos terms as sin.
Spoiler:
Expand the brackets
Spoiler:
Collect like terms and simplify
Spoiler:
Hello, RogueDemon!
We should work on one side only
. . and try to make it equal the other side.
14. a) Which equations are not identities? Justify your answers.
. . .b) For those equations that are identities, state any restrictions on the variables.
I started with the left side:
. . . . . . . . . . . .Replace with
. . . . . . . . . . . . . Replace with
. . . . . . . . . . . . Subtract the fractions
. . . . . . . . . . Replace with
. . . . . . . Combine like terms
. . . . . . . . . . . .Multiply
. . . . . . . . . . . . . . .Replace with
. . . . .