simplify the expression.
cot^2(x)-csc^2(-x)
csc(-x) = -csc(x)
sin^2+cos^2 = 1
But my teacher gave the answer on the review as -1.
but but but ... you should have used my hints.
csc^2(-x) = (-cscx)^2 = csc^2(x)
sin^2+cos^2 = 1
divide by sin^2 to get
1+cot^2 = csc^2
cot^2 - csc^2 = -1
sin^2+cos^2=1 will help solve many problems.
To simplify the expression cot^2(x) - csc^2(-x), let's start by expanding the trigonometric functions using their definitions:
cot^2(x) = (cos(x)/sin(x))^2
csc^2(-x) = (1/sin(-x))^2
Now, let's simplify each term separately:
cot^2(x) = (cos(x)/sin(x))^2
= cos^2(x)/sin^2(x)
csc^2(-x) = (1/sin(-x))^2
= 1/sin^2(-x)
Next, let's simplify the expression by substituting back the simplified forms:
cot^2(x) - csc^2(-x) = cos^2(x)/sin^2(x) - 1/sin^2(-x)
Since sin(-x) = -sin(x), we can substitute sin^2(-x) with sin^2(x):
cot^2(x) - csc^2(-x) = cos^2(x)/sin^2(x) - 1/sin^2(x)
Now, to simplify further, we need to find the common denominator for the fractions. The common denominator is sin^2(x):
cot^2(x) - csc^2(-x) = cos^2(x)/sin^2(x) - 1/sin^2(x)
To combine the fractions, we can subtract the numerators and place them over the common denominator:
cot^2(x) - csc^2(-x) = (cos^2(x) - 1)/sin^2(x)
Finally, we can simplify the numerator if needed:
cos^2(x) - 1 = -sin^2(x)
Therefore, the simplified expression is:
cot^2(x) - csc^2(-x) = -sin^2(x)/sin^2(x)
= -1