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Question
\frac{1}{\cot^{2}\left(a\right)}\times \frac{1}{1\times \tan^{2}\left(a\right)}=\frac{1}{1-\sin^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)}
Solve the equation
a \in \varnothing
Alternative Form
\textrm{No solution}
Evaluate
\frac{1}{\cot^{2}\left(a\right)}\times \frac{1}{1\times \tan^{2}\left(a\right)}=\frac{1}{1-\sin^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)}
Find the domain
More Steps
Evaluate
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\\cot^{2}\left(a\right)\neq 0\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\1\times \tan^{2}\left(a\right)\neq 0\\1-\sin^{2}\left(a\right)\neq 0\\\csc^{2}\left(a\right)\neq 0\end{array}\right.
Calculate
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\1\times \tan^{2}\left(a\right)\neq 0\\1-\sin^{2}\left(a\right)\neq 0\\\csc^{2}\left(a\right)\neq 0\end{array}\right.
Calculate
More Steps
Evaluate
1\times \tan^{2}\left(a\right)\neq 0
Any expression multiplied by 1 remains the same
\tan^{2}\left(a\right)\neq 0
Rewrite the expression
a\neq k\pi ,k \in \mathbb{Z}
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq k\pi ,k \in \mathbb{Z}\\1-\sin^{2}\left(a\right)\neq 0\\\csc^{2}\left(a\right)\neq 0\end{array}\right.
Calculate
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\csc^{2}\left(a\right)\neq 0\end{array}\right.
Calculate
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a \in \mathbb{R}\end{array}\right.
Simplify
\left\{ \begin{array}{l}a\neq k\pi ,k \in \mathbb{Z}\\a\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\a \in \mathbb{R}\end{array}\right.
Find the intersection
a\neq \frac{k\pi }{2},k \in \mathbb{Z}
\frac{1}{\cot^{2}\left(a\right)}\times \frac{1}{1\times \tan^{2}\left(a\right)}=\frac{1}{1-\sin^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)},a\neq \frac{k\pi }{2},k \in \mathbb{Z}
Simplify
More Steps
Evaluate
\frac{1}{\cot^{2}\left(a\right)}\times \frac{1}{1\times \tan^{2}\left(a\right)}
Reduce the fraction
\frac{1}{\cot^{2}\left(a\right)}\times \frac{1}{\tan^{2}\left(a\right)}
Multiply the terms
\frac{1}{\cot^{2}\left(a\right)\tan^{2}\left(a\right)}
Multiply the terms
\frac{1}{\left(\cot\left(a\right)\tan\left(a\right)\right)^{2}}
Rewrite the expression
\left(\cot\left(a\right)\tan\left(a\right)\right)^{-2}
Transform the expression
More Steps
Evaluate
\cot\left(a\right)\tan\left(a\right)
Calculate
\frac{\cos\left(a\right)}{\sin\left(a\right)}\times \tan\left(a\right)
Calculate
\frac{\cos\left(a\right)}{\sin\left(a\right)}\times \frac{\sin\left(a\right)}{\cos\left(a\right)}
Calculate
1
1^{-2}
Simplify
1
1=\frac{1}{1-\sin^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)}
Simplify
More Steps
Evaluate
\frac{1}{1-\sin^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)}
Calculate
\frac{1}{\cos^{2}\left(a\right)}-\frac{1}{\csc^{2}\left(a\right)}
Calculate
\frac{1}{\cos^{2}\left(a\right)}-\csc^{-2}\left(a\right)
Simplify
More Steps
Evaluate
-\csc^{-2}\left(a\right)
Transform the expression
-\left(\frac{1}{\sin\left(a\right)}\right)^{-2}
Simplify
-\sin^{2}\left(a\right)
\frac{1}{\cos^{2}\left(a\right)}-\sin^{2}\left(a\right)
Simplify
\cos^{-2}\left(a\right)-1+\cos^{2}\left(a\right)
Calculate
1+\tan^{2}\left(a\right)-1+\cos^{2}\left(a\right)
Calculate
\tan^{2}\left(a\right)+\cos^{2}\left(a\right)
1=\tan^{2}\left(a\right)+\cos^{2}\left(a\right)
Swap the sides of the equation
\tan^{2}\left(a\right)+\cos^{2}\left(a\right)=1
Rewrite the expression
\left(\frac{\sin\left(a\right)}{\cos\left(a\right)}\right)^{2}+\cos^{2}\left(a\right)=1
Simplify
\frac{\sin^{2}\left(a\right)}{\cos^{2}\left(a\right)}+\cos^{2}\left(a\right)=1
Multiply both sides of the equation by LCD
\left(\frac{\sin^{2}\left(a\right)}{\cos^{2}\left(a\right)}+\cos^{2}\left(a\right)\right)\cos^{2}\left(a\right)=1\times \cos^{2}\left(a\right)
Simplify the equation
More Steps
Evaluate
\left(\frac{\sin^{2}\left(a\right)}{\cos^{2}\left(a\right)}+\cos^{2}\left(a\right)\right)\cos^{2}\left(a\right)
Apply the distributive property
\frac{\sin^{2}\left(a\right)}{\cos^{2}\left(a\right)}\times \cos^{2}\left(a\right)+\cos^{2}\left(a\right)\cos^{2}\left(a\right)
Simplify
\sin^{2}\left(a\right)+\cos^{2}\left(a\right)\cos^{2}\left(a\right)
Calculate
More Steps
Multiply the terms
\cos^{2}\left(a\right)\cos^{2}\left(a\right)
Calculate
\cos^{2+2}\left(a\right)
Calculate
\cos^{4}\left(a\right)
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)=1\times \cos^{2}\left(a\right)
Any expression multiplied by 1 remains the same
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)=\cos^{2}\left(a\right)
Move the expression to the left side
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)-\cos^{2}\left(a\right)=0
Simplify
More Steps
Evaluate
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)-\cos^{2}\left(a\right)
Calculate
\sin^{2}\left(a\right)+\cos^{4}\left(a\right)-1+\sin^{2}\left(a\right)
Calculate
2\sin^{2}\left(a\right)+\cos^{4}\left(a\right)-1
2\sin^{2}\left(a\right)+\cos^{4}\left(a\right)-1=0
Simplify the equation using the Weierstrass substitution
2\left(\frac{2\tan\left(\frac{1}{2}a\right)}{1+\tan^{2}\left(\frac{1}{2}a\right)}\right)^{2}+\left(\frac{1-\tan^{2}\left(\frac{1}{2}a\right)}{1+\tan^{2}\left(\frac{1}{2}a\right)}\right)^{4}-1=0
Solve using substitution
2\left(\frac{2t}{1+t^{2}}\right)^{2}+\left(\frac{1-t^{2}}{1+t^{2}}\right)^{4}-1=0
Rearrange the terms
\begin{align}&2\left(\frac{2t}{1+t^{2}}\right)^{2}+\left(\frac{1-t^{2}}{1+t^{2}}\right)^{4}-1=0\\&a=180^{\circ}+360^{\circ} k,k \in \mathbb{Z}\end{align}
Calculate
More Steps
Evaluate
2\left(\frac{2t}{1+t^{2}}\right)^{2}+\left(\frac{1-t^{2}}{1+t^{2}}\right)^{4}-1=0
Calculate the sum or difference
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Evaluate
2\left(\frac{2t}{1+t^{2}}\right)^{2}+\left(\frac{1-t^{2}}{1+t^{2}}\right)^{4}-1
Rewrite the expression
2\left(\frac{2t}{1+t^{2}}\right)^{2}+\frac{1-4t^{2}+6t^{4}-4t^{6}+t^{8}}{1+4t^{2}+6t^{4}+4t^{6}+t^{8}}-1
Rewrite the expression
\frac{8t^{2}}{\left(1+t^{2}\right)^{2}}+\frac{1-4t^{2}+6t^{4}-4t^{6}+t^{8}}{1+4t^{2}+6t^{4}+4t^{6}+t^{8}}-1
Factor the expression
\frac{8t^{2}}{\left(1+t^{2}\right)^{2}}+\frac{1-4t^{2}+6t^{4}-4t^{6}+t^{8}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}-1
Reduce fractions to a common denominator
\frac{8t^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}+\frac{1-4t^{2}+6t^{4}-4t^{6}+t^{8}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}-\frac{\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}
Rewrite the expression
\frac{8t^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}+\frac{1-4t^{2}+6t^{4}-4t^{6}+t^{8}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}-\frac{\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
Write all numerators above the common denominator
\frac{8t^{2}\left(1+2t^{2}+t^{4}\right)+1-4t^{2}+6t^{4}-4t^{6}+t^{8}-\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
Multiply the terms
\frac{8t^{2}+16t^{4}+8t^{6}+1-4t^{2}+6t^{4}-4t^{6}+t^{8}-\left(1+t^{2}\right)^{2}\left(1+2t^{2}+t^{4}\right)}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
Multiply the terms
\frac{8t^{2}+16t^{4}+8t^{6}+1-4t^{2}+6t^{4}-4t^{6}+t^{8}-\left(1+t^{2}\right)^{4}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
Expand the expression
\frac{8t^{2}+16t^{4}+8t^{6}+1-4t^{2}+6t^{4}-4t^{6}+t^{8}-\left(1+4t^{2}+6t^{4}+4t^{6}+t^{8}\right)}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
Calculate the sum or difference
\frac{16t^{4}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}
\frac{16t^{4}}{\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}}=0
Cross multiply
16t^{4}=\left(1+2t^{2}+t^{4}\right)\left(1+t^{2}\right)^{2}\times 0
Simplify the equation
16t^{4}=0
Rewrite the expression
t^{4}=0
The only way a power can be 0 is when the base equals 0
t=0
\begin{align}&t=0\\&a=180^{\circ}+360^{\circ} k,k \in \mathbb{Z}\end{align}
Substitute back
\begin{align}&\tan\left(\frac{1}{2}a\right)=0\\&a=180^{\circ}+360^{\circ} k,k \in \mathbb{Z}\end{align}
Calculate
More Steps
Evaluate
\tan\left(\frac{1}{2}a\right)=0
Use the inverse trigonometric function
\frac{1}{2}a=\arctan\left(0\right)
Calculate
\frac{1}{2}a=0
\text{Add the period of }k\pi ,k \in \mathbb{Z}\text{ to find all solutions}
\frac{1}{2}a=k\pi ,k \in \mathbb{Z}
Solve the equation
More Steps
Evaluate
\frac{1}{2}a=k\pi
Multiply by the reciprocal
\frac{1}{2}a\times 2=k\pi \times 2
Multiply
a=k\pi \times 2
Multiply
a=2k\pi
a=2k\pi ,k \in \mathbb{Z}
\begin{align}&a=2k\pi ,k \in \mathbb{Z}\\&a=180^{\circ}+360^{\circ} k,k \in \mathbb{Z}\end{align}
\text{Check if }x=\pi+2k\pi,k\in\mathbb{Z}\text{ is a solution}
\frac{1}{\cot^{2}\left(\pi +2k\pi\right)}\times \frac{1}{1\times \tan^{2}\left(\pi +2k\pi\right)}=\frac{1}{1-\sin^{2}\left(\pi +2k\pi\right)}-\frac{1}{\csc^{2}\left(\pi +2k\pi\right)}
Calculate
\frac{1}{\cot^{2}\left(\pi \right)}\times \frac{1}{1\times \tan^{2}\left(\pi \right)}=\frac{1}{1-\sin^{2}\left(\pi \right)}-\frac{1}{\csc^{2}\left(\pi \right)}
Evaluate
\text{Undefined}
\text{Since }x=\pi+2k\pi,k\in\mathbb{Z}\text{ is not a solution,don't include it}
\begin{align}&a=2k\pi ,k \in \mathbb{Z}\\&a=180^{\circ}+360^{\circ} k,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{k\pi }{2},k \in \mathbb{Z}
Solution
a \in \varnothing
Alternative Form
\textrm{No solution}
Show Solutions
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