Help please? I'm getting really confused.
>>>csc^2A+sec^2A = sec^2A csc^2A
>>>1+{(tan^2A)/(1+secA)]
And the "1" is getting me frustrated xO
csc^2 + sec^2
1/sin^2 + 1/cos^2
(sin^2 + cos^2)/(sin^2 cos^2)
1/(sin^2 cos^2)
sec^2 csc^2
Not sure what the 2nd line is saying
It's not an identity, just an expression.
Oh sorry I forgot to put in secA so it should be 1+{(tan^2A)/(1+secA)] = secA
I understand that you're feeling confused, but don't worry! I'll break down both equations for you step by step.
Let's start with the first equation:
csc^2A + sec^2A = sec^2A * csc^2A
To simplify it, we'll start by using the reciprocal identities:
csc^2A = 1/sin^2A
sec^2A = 1/cos^2A
Now, let's substitute these identities into the equation:
(1/sin^2A) + (1/cos^2A) = (1/cos^2A) * (1/sin^2A)
Next, we'll find the common denominator for the left side of the equation:
[(1 * cos^2A) + (1 * sin^2A)] / (sin^2A * cos^2A) = (1/cos^2A) * (1/sin^2A)
Simplifying the numerator gives us:
(cos^2A + sin^2A) / (sin^2A * cos^2A) = (1/cos^2A) * (1/sin^2A)
Using the Pythagorean identity, cos^2A + sin^2A equals 1:
1 / (sin^2A * cos^2A) = (1/cos^2A) * (1/sin^2A)
Notice that both the numerator and denominator on each side of the equation are reciprocals of each other, so we can cancel them out. This leaves us with:
1 = 1
Therefore, the equation is true. Both sides of the equation are equal, and there are no restrictions on the values of A.
Now, let's move on to the second equation:
1 + (tan^2A / (1 + secA))
To simplify this equation, let's start by using the reciprocal identity for tan^2A:
tan^2A = (sin^2A / cos^2A)
Now, we can substitute this identity into the equation:
1 + ((sin^2A / cos^2A) / (1 + secA))
Next, let's find a common denominator for the fraction:
1 + ((sin^2A / cos^2A) * (1 / (1 + secA)))
Now, we need to simplify the expression by multiplying the fractions:
1 + ((sin^2A) / (cos^2A * (1 + secA)))
Next, we can simplify the denominator by using the identity secA = 1 / cosA:
1 + ((sin^2A) / (cos^2A * (1 + (1 / cosA))))
To combine the fractions, we need to find a common denominator:
1 + ((sin^2A) / ((cos^2A * (cosA + 1)) / cosA))
Simplifying the expression further:
1 + ((sin^2A * cosA) / (cos^3A + cos^2A))
Finally, let's combine the terms:
((cos^3A + cos^2A) + sin^2A * cosA) / (cos^3A + cos^2A)
This is the final form of the equation. Note that there are no restrictions on the values of A, so this equation holds for any value of A.
I hope this explanation helps clear up your confusion. If you have any further questions, feel free to ask!