transform the expression on left to one on right
csc x-sin x to cot x cos x
csc x-sin x = (1/sinx) - sin x
= (1 - sin^2 x)/sin x = (cos x * cos x)/sin x
= cos x cot x
We have trig identities.
cscx - sinx to cotx(cos x).
cscx is the reciprocal of the sine function.
So, then:
cscx = 1/sinx
We now have:
1/sinx - sinx...Treat this like a fraction case.
1/sinx - sinx becomes (1 - sin^2x)/sinx
Of course, (1 - sin^2x) is one of the Pythagorean Identities, recall?
We know that (1 - sin^2x) = cos^2x
We now have:
cos^2x/sinx = the right side
Done!
To transform the expression "csc x - sin x" into "cot x cos x," we'll make use of some trigonometric identities. Here's how you can do it step by step:
Step 1: Rewrite "csc x" as "1/sin x":
csc x - sin x = (1/sin x) - sinx
Step 2: Find the common denominator for subtraction:
To subtract the fractions, we need a common denominator. In this case, the common denominator is sin x. Multiply both terms by sin x:
= (1 - sin^2 x) / sin x
Step 3: Use the Pythagorean Identity for sin^2 x:
The Pythagorean Identity states that sin^2 x + cos^2 x = 1. Rearranging it, we have sin^2 x = 1 - cos^2 x. Substitute this value:
= (1 - (1 - cos^2 x)) / sin x
= (1 - 1 + cos^2 x) / sin x
= cos^2 x / sin x
Step 4: Use the identity cot x = cos x / sin x:
Replace cos^2 x / sin x with cot x cos x:
cos^2 x / sin x = (cos x / sin x) * cos x
= cot x cos x
Thus, the expression "csc x - sin x" can be transformed into "cot x cos x."