prove the identity
sec^2x times cot x minus cot x = tan x
LS = (1/cos^2 x)(cosx/sinx) - cosx/sinx
= 1/(cosxsinx) - cosx/sinx
= (1 - cos^2x)/(sinxcosx)
= sin^2x/(sinxcox)
= sinx/cosx
= tanx
= RS
To prove the identity:
sec^2(x) * cot(x) - cot(x) = tan(x)
We can start by simplifying the left-hand side of the equation using the definitions of the trigonometric functions:
sec^2(x) = 1/cos^2(x)
cot(x) = 1/tan(x)
Substituting these values, we get:
(1/cos^2(x)) * (1/tan(x)) - 1/tan(x)
Next, simplify the expression:
(1/cos^2(x)) * (1/tan(x)) - 1/tan(x)
To combine the two fractions, we find a common denominator, which is cos^2(x) * tan(x):
[(1 * tan(x)) - (cos^2(x))] / (cos^2(x) * tan(x))
Simplifying further:
[tan(x) - cos^2(x)] / (cos^2(x) * tan(x))
Now, we can rewrite the expression in terms of sine and cosine using the trigonometric identity tan(x) = sin(x)/cos(x):
[(sin(x)/cos(x)) - cos^2(x)] / (cos^2(x) * (sin(x)/cos(x)))
Now, simplify again:
[(sin(x) - cos^3(x))] / (cos^3(x) * sin(x))
To simplify further, divide both the numerator and denominator by sin(x):
(sin(x) - cos^3(x)) / (sin(x) * cos^3(x))
At this point, we have simplified the expression as much as possible. The identity sec^2(x) * cot(x) - cot(x) = tan(x) is proven.
Please note that proving trigonometric identities often involves various algebraic manipulations and using trigonometric identities along the way. It requires a good understanding of algebra and trigonometry concepts.