Simplify the expressions
1.) Cos^2x/sin^2x + csc x sin x
2.) (Sec x +1)(sec x - 1)
cot = cos/sin
csc = 1/sin
so, cos^2/sin^2 + csc*sin
= cot^2 + 1
= csc^2
(sec+1)(sec-1) = sec^2 - 1 = tan^2
To simplify these expressions, we can use trigonometric identities. Let's break it down step by step:
1.) Cos^2x/sin^2x + csc x sin x:
First, let's simplify the expression cos^2x/sin^2x. Using the identity sin^2x + cos^2x = 1, we can rewrite cos^2x as 1 - sin^2x.
So, the expression becomes (1 - sin^2x)/sin^2x + csc x sin x.
Now, let's find a common denominator for the two terms. The common denominator is sin^2x.
The expression becomes (1 - sin^2x + csc x sin^3x)/sin^2x.
Next, let's simplify the second term csc x sin^3x. Using the identity csc x = 1/sin x, we can rewrite it as (1/sin x) * sin^3x = sin^2x.
Therefore, the expression simplifies to (1 - sin^2x + sin^2x)/sin^2x, which further simplifies to (1/sin^2x), or simply csc^2x.
2.) (Sec x +1)(Sec x - 1):
Let's multiply these two expressions using the distributive property. We multiply Sec x with each term in the second parentheses:
Sec x * Sec x - Sec x * 1 + 1 * Sec x - 1 * 1
This becomes Sec^2x - Sec x + Sec x - 1.
Note that the middle terms Sec x and -Sec x cancel each other out, so the expression simplifies to:
Sec^2x - 1.
And there you have it, the simplified expression is Sec^2x - 1.