**Sin/1-cos**

To simplify the expression Sin/1-cos, we can use the trigonometric identity (1-cos^2 x) = sin^2 x. By substituting this identity into the expression, we get:

Sin/1-cos = Sin/(1-cos) * (1+cos)/(1+cos)
= Sin(1+cos)/((1-cos)(1+cos))
= Sin(1+cos)/(1-cos^2)

Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can substitute sin^2 x with (1-cos^2 x):

Sin/1-cos = Sin(1+cos)/(1-cos^2)
= Sin(1+cos)/(sin^2 x)

Since sin^2 x is equal to 1-cos^2 x, we can further simplify the expression:

Sin/1-cos = Sin(1+cos)/(sin^2 x)
= Sin(1+cos)/(1-cos^2 x)

**tan/1**

The expression tan/1 can be simplified as tan x/1. Since tan x = sin x / cos x, we can substitute tan x with sin x / cos x:

tan/1 = (sin x / cos x) / 1
= sin x / cos x

**cos = sec × cosec cot**

To prove the trigonometric identity cos x = sec x × cosec x cot x, we need to manipulate the expression using other trigonometric identities.

Starting with the right-hand side of the equation:

sec x × cosec x cot x
= (1 / cos x) × (1 / sin x) × (cos x / sin x)
= 1 / (sin x × sin x)
= 1 / sin^2 x

Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can substitute sin^2 x with (1 - cos^2 x):

1 / sin^2 x
= 1 / (1 - cos^2 x)

Finally, using the reciprocal identity sec x = 1 / cos x, we can rewrite the expression:

1 / (1 - cos^2 x)
= 1 / (sin^2 x)
= cos x

Therefore, we have proven that cos x = sec x × cosec x cot x.