Final answer:
To factor the expression sin²x+2/cscx+1, rewrite it in terms of sine and cosine functions. Simplify the expression by multiplying the numerator and denominator of the fraction by sin(x). Finally, factor out a common factor of sin(x) from all three terms.
Step-by-step explanation:
To factor the expression sin²x+2/cscx+1, we can rewrite it using trigonometric identities. First, we rewrite the expression in terms of sine and cosine functions. Since csc(x) is the reciprocal of sin(x), we can replace csc(x) with 1/sin(x). So, the expression becomes:
sin²x + 2/(1/sin(x)) + 1
Next, we simplify the expression by multiplying the numerator and denominator of the fraction by sin(x) to get:
sin²x + 2sin(x) + sin(x)
Finally, we can factor out a common factor of sin(x) from all three terms:
sin(x)(sin(x) + 2 + 1)
Therefore, the factored form of the expression sin²x+2/cscx+1 is sin(x)(sin(x) + 3).