For csc^2 A-1/cot A csc A, what is the simplest equivalent trig expression? The answers I have to choose from are sin A, cosA, tan A or csc A, but I don't know how?? csc theta= 1/sin theta but its ^2. Help please I don't even no where to begin
csc=1/sin
cot=cos/sin
so,
csc^2-1/cotcsc =
((1/sin^2)-1)sin/(cos/sin) =
sin((sin/sin^2)-sin)/(cos) =
(sin^2/sin^2-sin^2)/cos =
(1-sin^2)/cos =
cos^2/cos = cos A
((sin^2/sin^2)-sin/(sin^2 cos)
To simplify the expression (csc^2 A - 1) / (cot A * csc A), we need to use trigonometric identities.
Let's break it down step by step:
Step 1: Replace csc^2 A with (1/sin^2 A)
The reciprocal of the trigonometric function csc A is 1/sin A. Since csc A is squared in the expression, its reciprocal becomes (1/sin A)^2, which simplifies to 1/sin^2 A.
Now our expression is: (1/sin^2 A - 1) / (cot A * csc A)
Step 2: Combine the fractions
To combine the fractions, we need a common denominator. The common denominator of sin^2 A and 1 is sin^2 A. To make the fractions have a common denominator, we can multiply the numerator and denominator of 1 by sin^2 A:
[(1 - sin^2 A) / sin^2 A] / (cot A * csc A)
Step 3: Simplify the numerator
Using the identity sin^2 A + cos^2 A = 1, we can rewrite 1 - sin^2 A as cos^2 A. Therefore, the numerator simplifies to cos^2 A:
[cos^2 A / sin^2 A] / (cot A * csc A)
Step 4: Simplify the denominator
Recall that cot A = 1/tan A and csc A = 1/sin A. Substituting these identities, the denominator becomes (1/tan A) * (1/sin A), which can be simplified to 1/(tan A * sin A):
[cos^2 A / sin^2 A] / (1/(tan A * sin A))
Step 5: Invert and multiply
When dividing by a fraction, we can invert the second fraction and multiply. Therefore, we have:
[cos^2 A / sin^2 A] * (tan A * sin A) / 1
Step 6: Cancel out common factors
Let's cancel out the common factors between the numerator and denominator. Notice that sin A appears in both the numerator and denominator, so it can be canceled:
(cos^2 A * tan A) / 1
Step 7: Simplify the expression
Finally, we have:
cos^2 A * tan A
Now we can see that the simplified expression is cosine squared (cos^2 A) times tangent (tan A).
Out of the options sin A, cos A, tan A, and csc A, the simplest equivalent trig expression is tan A.