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Question
\sec\left(a\right)-\frac{1}{\sec\left(a\right)}+1=1-\frac{\cos\left(a\right)}{1}+\cos\left(a\right)
Solve the equation
a=k\pi ,k \in \mathbb{Z}
Alternative Form
a=180^{\circ} k,k \in \mathbb{Z}
Evaluate
\sec\left(a\right)-\frac{1}{\sec\left(a\right)}+1=1-\frac{\cos\left(a\right)}{1}+\cos\left(a\right)
Find the domain
More Steps
Evaluate
\left\{ \begin{array}{l}a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\sec\left(a\right)\neq 0\end{array}\right.
Calculate
\left\{ \begin{array}{l}a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a \in \mathbb{R}\end{array}\right.
Find the intersection
a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
\sec\left(a\right)-\frac{1}{\sec\left(a\right)}+1=1-\frac{\cos\left(a\right)}{1}+\cos\left(a\right),a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Cancel equal terms on both sides of the expression
\sec\left(a\right)-\frac{1}{\sec\left(a\right)}=-\frac{\cos\left(a\right)}{1}+\cos\left(a\right)
Simplify
More Steps
Evaluate
-\frac{\cos\left(a\right)}{1}+\cos\left(a\right)
Divide the terms
-\cos\left(a\right)+\cos\left(a\right)
Transform the expression
\cos\left(a\right)-\cos\left(a\right)
Calculate
0
\sec\left(a\right)-\frac{1}{\sec\left(a\right)}=0
Multiply both sides of the equation by LCD
\left(\sec\left(a\right)-\frac{1}{\sec\left(a\right)}\right)\sec\left(a\right)=0\times \sec\left(a\right)
Simplify the equation
More Steps
Evaluate
\left(\sec\left(a\right)-\frac{1}{\sec\left(a\right)}\right)\sec\left(a\right)
Apply the distributive property
\sec\left(a\right)\sec\left(a\right)-\frac{1}{\sec\left(a\right)}\times \sec\left(a\right)
Simplify
\sec\left(a\right)\sec\left(a\right)-1
Multiply the terms
\sec^{2}\left(a\right)-1
\sec^{2}\left(a\right)-1=0\times \sec\left(a\right)
Any expression multiplied by 0 equals 0
\sec^{2}\left(a\right)-1=0
Add or subtract both sides
\sec^{2}\left(a\right)=0+1
Removing 0 doesn't change the value,so remove it from the expression
\sec^{2}\left(a\right)=1
Take the root of both sides of the equation and remember to use both positive and negative roots
\sec\left(a\right)=\pm \sqrt{1}
Simplify the expression
\sec\left(a\right)=\pm 1
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&\sec\left(a\right)=1\\&\sec\left(a\right)=-1\end{align}
Calculate
More Steps
Evaluate
\sec\left(a\right)=1
Use the inverse trigonometric function
a=\operatorname{arcsec}\left(1\right)
Calculate
a=0
\text{Add the period of }2k\pi ,k \in \mathbb{Z}\text{ to find all solutions}
a=2k\pi ,k \in \mathbb{Z}
\begin{align}&a=2k\pi ,k \in \mathbb{Z}\\&\sec\left(a\right)=-1\end{align}
Calculate
More Steps
Evaluate
\sec\left(a\right)=-1
Use the inverse trigonometric function
a=\operatorname{arcsec}\left(-1\right)
Calculate
a=\pi
\text{Add the period of }2k\pi ,k \in \mathbb{Z}\text{ to find all solutions}
a=\pi +2k\pi ,k \in \mathbb{Z}
\begin{align}&a=2k\pi ,k \in \mathbb{Z}\\&a=\pi +2k\pi ,k \in \mathbb{Z}\end{align}
Find the union
a=k\pi ,k \in \mathbb{Z}
Check if the solution is in the defined range
a=k\pi ,k \in \mathbb{Z},a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Solution
a=k\pi ,k \in \mathbb{Z}
Alternative Form
a=180^{\circ} k,k \in \mathbb{Z}
Show Solutions
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