1. (1-sin²B)/(sin²Bcos²B)= csc²B-sec²B
--that's really the given. please help.
The left side is
cos^2 B/(sin^2 B*cos^2 B) = 1/sin^2 B
= csc^2 B
That does not equal the right side unless sec B = 0, which is not possible. It is not a valid identity.
You can prove that to yourself by picking any angle B. Let's take 30 degrees.
sin^2 30 = 1/4
cos^2 30 = 3/4
csc^2 30 = 4
sec^2 30 = 4/3
(3/4)/[3/16] = 4 is the left side
4 - 4/3 = 5/3 is the right side
Obviously my last calculation is wrong and should be
4 - 4/3 = 8/3
In any case, the "identity" is not valid
To prove the given equation:
Start with the left side of the equation:
(1 - sin²B) / (sin²B cos²B)
Using the identity sin²B = 1 - cos²B, we can rewrite the numerator:
= cos²B / (sin²B cos²B)
Cancel out the common factor of cos²B:
= 1/sin²B
Using the reciprocal identities, we can rewrite 1/sin²B as csc²B:
= csc²B
Now let's move to the right side of the equation:
csc²B - sec²B
Using the definitions of cscB and secB:
= (1/sinB)² - (1/cosB)²
Using the reciprocal identities:
= (1/sinB)² - (cosB/sinB)²
Using the exponent rules:
= (1 - cos²B) / sin²B
Using the identity sin²B = 1 - cos²B, we have:
= 1 / sin²B
Therefore, we have proven that:
(1 - sin²B) / (sin²B cos²B) = csc²B - sec²B