# Solve the inequality (sin(x))/(cos(2x+pi/2))%(-1sec(x))/2

## \frac{\sin\left(x\right)}{\cos\left(2x+\frac{\pi}{2}\right)}\geq \frac{-\sec\left(x\right)}{2}

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$\frac{\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}+\frac{1}{\csc\left(x\right)}\geq 0$

## Step by step solution

Problem

$\frac{\sin\left(x\right)}{\cos\left(2x+\frac{\pi}{2}\right)}\geq \frac{-\sec\left(x\right)}{2}$
1

Simplifying the fraction

$\frac{\sin\left(x\right)}{\cos\left(\frac{\pi}{2}+2x\right)}\geq -\frac{1}{2}\sec\left(x\right)$
2

Moving the denominator multiplying to the other side of the inequation

$\sin\left(x\right)\geq -\frac{1}{2}\cos\left(\sqrt{2}+2x\right)\sec\left(x\right)$
3

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\sin\left(x\right)\geq -\frac{1}{2}\cos\left(\sqrt{2}+2x\right)\frac{1}{\cos\left(x\right)}$
4

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{1}{2}$ and $x=\cos\left(x\right)$

$\sin\left(x\right)\geq \cos\left(\sqrt{2}+2x\right)\frac{-\frac{1}{2}}{\cos\left(x\right)}$
5

Multiplying the fraction and term

$\sin\left(x\right)\geq \frac{-\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}$
6

Applying the sine identity: $\displaystyle\sin\left(\theta\right)=\frac{1}{\csc\left(\theta\right)}$

$\frac{1}{\csc\left(x\right)}\geq \frac{-\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}$
7

Subtract $\frac{-\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}$ from both sides

$\frac{\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}+\frac{1}{\csc\left(x\right)}\geq 0$
8

Unifying fractions

$\frac{\frac{1}{2}\csc\left(x\right)\cos\left(\sqrt{2}+2x\right)+\cos\left(x\right)}{\cos\left(x\right)\csc\left(x\right)}\geq 0$
9

Split the fraction $\frac{\cos\left(x\right)+\frac{1}{2}\csc\left(x\right)\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)\csc\left(x\right)}$ in two terms with same denominator

$\frac{\frac{1}{2}\csc\left(x\right)\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)\csc\left(x\right)}+\frac{\cos\left(x\right)}{\cos\left(x\right)\csc\left(x\right)}\geq 0$
10

Simplifying the fraction by $\cos\left(x\right)$

$\frac{\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{1\cos\left(x\right)}+\frac{1}{\csc\left(x\right)}\geq 0$
11

Any expression multiplied by $1$ is equal to itself

$\frac{\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}+\frac{1}{\csc\left(x\right)}\geq 0$

$\frac{\frac{1}{2}\cos\left(\sqrt{2}+2x\right)}{\cos\left(x\right)}+\frac{1}{\csc\left(x\right)}\geq 0$

Inequalities

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