Which expression is equivalent to csc x - sin x?
To find an expression that is equivalent to csc(x) - sin(x), first, we need to find a common denominator for the two terms.
The reciprocal identity for sine is csc(x) = 1/sin(x). So, we can rewrite csc(x) as 1/sin(x).
Now, the expression becomes 1/sin(x) - sin(x).
To find a common denominator, we multiply the first term by sin(x)/sin(x):
1/sin(x) * sin(x)/sin(x) - sin(x) * sin(x)/sin(x)
This simplifies to:
sin(x)/sin(x)^2 - sin^2(x)/sin(x).
The common denominator is sin(x)^2. Now, we can combine the fractions:
sin(x) - sin^2(x) / sin(x)^2.
Therefore, an expression equivalent to csc(x) - sin(x) is (sin(x) - sin^2(x)) / sin(x)^2.
To find an expression equivalent to csc x - sin x, we need to apply trigonometric identities.
Recall the reciprocal identity for cosecant: csc x = 1/sin x.
So, we can rewrite the expression as: 1/sin x - sin x.
To combine the terms, we need to find a common denominator. The common denominator here is sin x, which means we need to multiply both terms by sin x:
(1/sin x) * (sin x/sin x) - (sin x * sin x)
Simplifying this expression, we get:
1 - sin^2 x.
Therefore, the expression equivalent to csc x - sin x is 1 - sin^2 x.
Why are you switching names ?
Please stick with one name.
csc x - sin x
= 1/sinx - sinx
= (1 - sin^2 x)/sinx
= cos^2 x /sinx
= (cosx/sinx) /sinx
= cotx cscx