Hello! I'm having problems with these two math problems. I've tried to solve them but I got nowhere. If someone could please help me, that would great! Thanks again! :)
Directions: Verify the identity for each problem.
1.) cot^2y(sec^2-y-1)=1
2.) cosx+sin xtan x=sec x
The expression sec^2-y-1 makes no sense to me
is it sec^2 (-y-1) ?
#2
LS = cosx + sinx(sinx/cosx)
= (cos^2 x + sin^2 x)/cosx
= 1/cosx
= sec x
= RS
I think that's just sec^2 y - 1, which is tan^2 y
cot^2 y * tan^2 y = 1
Hello! I'd be happy to help you with your math problems.
Let's start with the first problem:
1.) cot^2y(sec^2 y - 1) = 1
To verify this identity, we need to simplify both sides of the equation and see if they are equal.
On the left side, we have cot^2y multiplied by (sec^2 y - 1). We can start by simplifying the square of cot y:
cot^2y = (cos^2 y) / (sin^2 y)
Now let's simplify the expression inside the parentheses:
(sec^2 y - 1) = (1/cos^2 y) - 1
To simplify further, combine the fractions:
(1/cos^2 y) - 1 = (1 - cos^2 y) / cos^2 y
Next, substitute the simplified expressions back into the original equation:
(cos^2 y) / (sin^2 y) * ((1 - cos^2 y) / cos^2 y) = 1
To simplify further, we cancel out common terms:
(cos^2 y) * (1 - cos^2 y) / (sin^2 y * cos^2 y) = 1
Now we can simplify and see if both sides of the equation are equal. Expand the numerator:
((cos^2 y) - (cos^4 y)) / (sin^2 y * cos^2 y) = 1
Let's simplify the denominator by using the identity: sin^2 y = 1 - cos^2 y
(1 - cos^4 y) / (cos^2 y * (1 - cos^2 y)) = 1
Now, cancel out common terms:
(1 - cos^4 y) / (cos^2 y - (cos^4 y)) = 1
And simplify:
1/(cos^2 y) = 1
This is true because the reciprocal of the square of cos y is indeed equal to 1.
Therefore, we have verified the identity for the first problem.
Let's move on to the second problem:
2.) cos x + sin x * tan x = sec x
To verify this identity, we need to simplify both sides of the equation and see if they are equal.
On the left side, we have cos x plus sin x multiplied by tan x.
Let's simplify the expression inside the parentheses:
tan x = sin x / cos x
Now substitute the simplified expression back into the original equation:
cos x + sin x * (sin x / cos x) = sec x
Expand the expression:
cos x + sin^2 x / cos x = sec x
To simplify further, let's find a common denominator, which is cos x:
(cos x * cos x / cos x) + sin^2 x / cos x = sec x
Combine the terms:
(cos^2 x + sin^2 x) / cos x = sec x
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can simplify further:
1 / cos x = sec x
This is true because the reciprocal of cos x is indeed equal to sec x.
Therefore, we have verified the identity for the second problem.
I hope this explanation helps you understand how to verify identities in math problems. If you have any further questions, feel free to ask!