To factor the expression 1 – csc²(x) in terms of a single trigonometric function, we first need to remember the relationship between the cosecant function and the sine function. The cosecant function is defined as:
csc(x) = 1/sin(x)
Therefore, we can write:
csc²(x) = 1/sin²(x)
Substituting this into our expression gives:
1 – csc²(x) = 1 – (1/sin²(x))
To simplify this expression, we need a common denominator:
1 can be rewritten as sin²(x)/sin²(x). Now the expression becomes:
1 – csc²(x) = (sin²(x)/sin²(x)) – (1/sin²(x))
Now, we have:
1 – csc²(x) = (sin²(x) – 1) / sin²(x)
We can factor the numerator, sin²(x) – 1, which is a difference of squares:
sin²(x) – 1 = (sin(x) – 1)(sin(x) + 1)
This leads us to:
1 – csc²(x) = ((sin(x) – 1)(sin(x) + 1)) / sin²(x)
Thus, we have factored the algebraic expression in terms of sine, a single trigonometric function:
1 – csc²(x) = -cot²(x)
So the final factored form is:
1 – csc²(x) = -cot²(x)
This result is useful not only for simplifying calculations but also for solving problems involving trigonometric identities.