Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 19, 2023

Simplify the expression $\frac{8}{\cot^2(x) + 1}$ using fundamental identities. Verify the result numerically using a graphing utility.

STEP 1

Assumptions1. We are given the expression $\frac{8}{\cot ^{}(x)+1}$ . . We need to simplify this expression using the fundamental trigonometric identities.
3. The fundamental identities include the Pythagorean identities, reciprocal identities, quotient identities, and co-function identities.

STEP 2

We can use the Pythagorean identity $\cot ^{2}(x) +1 = \csc ^{2}(x)$ to simplify the denominator of the given expression.

STEP 3

Substitute $\csc ^{2}(x)$ for $\cot ^{2}(x) +1$ in the given expression.
$\frac{8}{\cot ^{2}(x)+1} = \frac{8}{\csc ^{2}(x)}$

STEP 4

Now, we can use the reciprocal identity $\csc(x) = \frac{1}{\sin(x)}$ to further simplify the expression.

STEP 5

Substitute $\frac{1}{\sin(x)}$ for $\csc(x)$ in the expression.
$\frac{8}{\csc ^{2}(x)} = \frac{8}{\left(\frac{1}{\sin(x)}\right)^2}$

STEP 6

implify the expression by multiplying the numerator and denominator by $\sin^2(x)$ .
$\frac{8}{\left(\frac{1}{\sin(x)}\right)^2} = \frac{8\sin^2(x)}{1} =8\sin^2(x)$ So, the simplified form of the given expression is $8\sin^2(x)$ .

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