How do you solve this problem:
sin(90* - x)
-----------------
cot^2(90* - x) + 1
The * is a degree symbol!!
I haven't seen any problems with degrees in it, usually it's only a variable! If anyone could give me guidelines or show me where to start, I'd really appreciate it!
THANKS!!!
To solve this problem, we first need to understand the trigonometric relationships involved and the properties of the trigonometric functions.
Let's start by simplifying the expression:
sin(90° - x) / (cot^2(90° - x) + 1)
To simplify this, we can use the identities:
sin(90° - x) = cos(x) (identity: sin(90° - x) = cos(x))
cot(90° - x) = tan(x) (identity: cot(90° - x) = tan(x))
cot^2(90° - x) = tan^2(x) = 1/cos^2(x) (identity: cot^2(90° - x) = tan^2(x) = 1/cos^2(x))
cot^2(90° - x) + 1 = 1/cos^2(x) + 1 = (1 + cos^2(x))/cos^2(x) = sec^2(x)/cos^2(x) (identity: cot^2(90° - x) + 1 = sec^2(x)/cos^2(x))
Now, we can substitute these simplifications back into the expression:
cos(x) / (sec^2(x)/cos^2(x))
Next, we can simplify further by multiplying the expression by the reciprocal of sec^2(x)/cos^2(x), which is cos^2(x)/sec^2(x):
(cos(x) / (sec^2(x)/cos^2(x))) * (cos^2(x)/sec^2(x))
This simplifies to:
cos^3(x) / sec^2(x)
Recall that sec(x) is equal to 1/cos(x). Substituting this in:
cos^3(x) / (1/cos^2(x))
To simplify further, we multiply the numerator by cos^2(x) and the denominator by cos^3(x):
cos^5(x) / (1)
Finally, the simplified expression is:
cos^5(x)
So, the solution to the problem is cos^5(x).
To summarize the steps:
1. Use trigonometric identities to simplify the expression.
2. Substitute the simplified expressions back into the original expression.
3. Simplify further using reciprocal and multiplication properties.
4. Apply trigonometric identity to simplify.
5. Simplify the expression to the final solution.
Remember to verify the solution and check for any domain restrictions on the value of x.