\sin(x)-\cos(x)=0
Question
\sin\left(x\right)-\cos\left(x\right)=0
Solve the equation
\text{Solve for }x
x=\frac{\pi }{4}+k\pi ,k \in \mathbb{Z}
Alternative Form
x=45^{\circ}+180^{\circ} k,k \in \mathbb{Z}
Alternative Form
x\approx 0.785398+k\pi ,k \in \mathbb{Z}
Evaluate
\sin\left(x\right)-\cos\left(x\right)=0
Rearrange the terms
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Simplify
\frac{2\tan\left(\frac{x}{2}\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}-\frac{1-\tan^{2}\left(\frac{x}{2}\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}
Add the terms
\frac{2\tan\left(\frac{x}{2}\right)-\left(1-\tan^{2}\left(\frac{x}{2}\right)\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}
Calculate
\frac{2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}
\frac{2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}=0
Multiply both sides of the equation by LCD
\frac{2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)}{1+\tan^{2}\left(\frac{x}{2}\right)}\times \left(1+\tan^{2}\left(\frac{x}{2}\right)\right)=0\times \left(1+\tan^{2}\left(\frac{x}{2}\right)\right)
Evaluate
2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)=0\times \left(1+\tan^{2}\left(\frac{x}{2}\right)\right)
Evaluate
2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)=0
Move the constant to the right side
2\tan\left(\frac{x}{2}\right)-1+\tan^{2}\left(\frac{x}{2}\right)-\left(-1\right)=0-\left(-1\right)
Add the terms
2\tan\left(\frac{x}{2}\right)+\tan^{2}\left(\frac{x}{2}\right)=1
Evaluate
\tan^{2}\left(\frac{x}{2}\right)+2\tan\left(\frac{x}{2}\right)=1
Add the same value to both sides
\tan^{2}\left(\frac{x}{2}\right)+2\tan\left(\frac{x}{2}\right)+1=1+1
Evaluate
\tan^{2}\left(\frac{x}{2}\right)+2\tan\left(\frac{x}{2}\right)+1=2
Evaluate
\left(\tan\left(\frac{x}{2}\right)+1\right)^{2}=2
Take the root of both sides of the equation and remember to use both positive and negative roots
\tan\left(\frac{x}{2}\right)+1=\pm \sqrt{2}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&\tan\left(\frac{x}{2}\right)+1=\sqrt{2}\\&\tan\left(\frac{x}{2}\right)+1=-\sqrt{2}\end{align}
Calculate
\begin{align}&\tan\left(\frac{x}{2}\right)=\sqrt{2}-1\\&\tan\left(\frac{x}{2}\right)+1=-\sqrt{2}\end{align}
Calculate
\begin{align}&\tan\left(\frac{x}{2}\right)=\sqrt{2}-1\\&\tan\left(\frac{x}{2}\right)=-\sqrt{2}-1\end{align}
Calculate
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Evaluate
\tan\left(\frac{x}{2}\right)=\sqrt{2}-1
Calculate
\frac{x}{2}=\arctan\left(\sqrt{2}-1\right)
Calculate
x=\frac{\pi }{4}+2k\pi ,k \in \mathbb{Z}
\begin{align}&x=\frac{\pi }{4}+2k\pi ,k \in \mathbb{Z}\\&\tan\left(\frac{x}{2}\right)=-\sqrt{2}-1\end{align}
Calculate
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Evaluate
\tan\left(\frac{x}{2}\right)=-\sqrt{2}-1
Calculate
\frac{x}{2}=\arctan\left(-\sqrt{2}-1\right)
Calculate
x=\frac{5\pi }{4}+2k\pi ,k \in \mathbb{Z}
\begin{align}&x=\frac{\pi }{4}+2k\pi ,k \in \mathbb{Z}\\&x=\frac{5\pi }{4}+2k\pi ,k \in \mathbb{Z}\end{align}
Solution
x=\frac{\pi }{4}+k\pi ,k \in \mathbb{Z}
Alternative Form
x=45^{\circ}+180^{\circ} k,k \in \mathbb{Z}
Alternative Form
x\approx 0.785398+k\pi ,k \in \mathbb{Z}
Graph