can someone please show me how to verify this identity?

(sec x/csc x minus cot x) minus (sec x/csc x + cot x)= 2 csc x

Is it

(sec x/(csc x - cot x) - (sec x/(csc x + cot x))= 2 csc x or
sec x/csc x - cot x - sec x/csc x + cot x= 2 csc x

The way you typed it, according to order of operation, it would be that last one, but that is not an identity.

so
(sec x/(csc x - cot x) - (sec x/(csc x + cot x))= 2 csc x

LS = (secxcscx + secxcotx - secxcscx + secxcotx)/(csc^2x - cot^2x)
= 2secxcotx/(csc^2x - (csc^2x - 1))
= 2(1/cosx)(cosx/sinx)
= 2/sinx
= 2cscx
= RS

Need some help with this: find all solutions of cot^2 x=3 in the interval [0,2pi)

Verify the Identity for (1-tanx)^2 = sec^2x-2tanx

To verify the given identity (sec x/csc x - cot x) - (sec x/csc x + cot x) = 2csc x, we need to simplify both sides of the equation and determine if they are equal.

Let's start by simplifying the left-hand side (LHS) of the equation:

(sec x/csc x - cot x) - (sec x/csc x + cot x)

To combine fractions, we need to find a common denominator. The common denominator for csc x and sec x is csc x. Therefore, we rewrite the equation as:

(sec x - cot x * csc x) - (sec x + cot x * csc x) / csc x

To simplify further, we can distribute the multiplication of cot x and csc x:

(sec x - cot x * csc x) - (sec x + cot x * csc x)
= (sec x - csc x / sin x) - (sec x + csc x / sin x)

Now, let's combine like terms:

(sec x - csc x / sin x) - (sec x + csc x / sin x)
= sec x - csc x / sin x - sec x - csc x / sin x

Next, we can simplify by combining the terms with sec x and csc x:

sec x - csc x / sin x - sec x - csc x / sin x
= -2csc x / sin x

Since sin x is the reciprocal of csc x, we can simplify further:

-2csc x / sin x
= -2csc x * sin x / 1
= -2

Therefore, the simplified LHS is -2.

Now, let's simplify the right-hand side (RHS) of the equation:

2csc x

We don't need to simplify this further as it is already in its simplest form.

Now, let's compare the simplified LHS and RHS:

LHS = -2
RHS = 2csc x

Since -2 is not equal to 2csc x, the given identity is not true.

To summarize, the given identity (sec x/csc x - cot x) - (sec x/csc x + cot x) = 2csc x is not verified as it is not true.