Sec Csc Problem

Question

Which expression is not equal to the following:

sec^2(x)csc^2(x)

A. sec^2(x) + csc^2(x)

B. cos^2(x) + sin^2(x)

C. sec^2(x) + sec^2(x) cot^2(x)

D. (1+ tan^2(x)) csc^2(x)

Answer 1

This question is easiest to visualize if we change everything to sine and cosine:

choice A:

1/cos²x + 1/sin²x

To add these, we need the least common denominator, which is sin²(x)cos²(x)

Multiply the first fraction by sin²x/sin²x and the second fraction by cos²x/cos²x

The new expression is sin²x/(sin²x cos²x) + cos²x/ (sin²x cos²x).

Simplify into one fraction and get (sin²x + cos²x)/ (sin²x cos²x). The numerator is an identity (sin²x + cos²x =1). We can substitute that identity in the numerator and be left with 1/(sin²x cos²x), which is equivalent to the original expression using reciprocal identities.

choice B:

This is an identity: (sin²x + cos²x =1)

choice C:

This can be rewritten as 1/cos²x + (1/cos²x)(cos²x/sin²x).

The cos²x can cancel in the second fraction, so the expression will simplify to 1/cos²x + 1/sin²x. Then it is the same as choice A.

choice D:

This can be rewritten as (1+ sin²x/cos²x)(1/sin²x). If we distribute the (1/sin²x), we get (1/sin²x)+(sin²x/(sin²xcos²x)). The sin²x will cancel in the second fraction, leaving us with 1/sin²x + 1/cos²x. Then, it is the same as choices A and C.

Answer 2

A good way of doing this is to just graph them all and look for the one that is unique. You can also look to see if anything stands out in the answer choices. One thing that is fairly easy to spot is that answer B is a Pythagorean Identity ( cos^2(x) + sin^2(x) = 1). The others are not equal to 1