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  • Equation:
  • Equation (x-4)/(x-6)=2 Equation (x-4)/(x-6)=2
  • Equation e^x=2 Equation e^x=2
  • Equation x/3+x/4=-21 Equation x/3+x/4=-21
  • Equation x^2+4*x=21 Equation x^2+4*x=21
  • Express {x} in terms of y where:
  • 19*x-3*y=-12
  • 17*x-7*y=15
  • 14*x+3*y=-4
  • 14*x-4*y=14

  • Identical expressions

  • (sina)^ four /(sinx)^ two +(cosa)^ four /(cosx)^ two = one
  • ( sinus of a) to the power of 4 divide by ( sinus of x) squared plus ( co sinus of e of a) to the power of 4 divide by ( co sinus of e of x) squared equally 1
  • ( sinus of a) to the power of four divide by ( sinus of x) to the power of two plus ( co sinus of e of a) to the power of four divide by ( co sinus of e of x) to the power of two equally one
  • (sina)4/(sinx)2+(cosa)4/(cosx)2=1
  • sina4/sinx2+cosa4/cosx2=1
  • (sina)⁴/(sinx)²+(cosa)⁴/(cosx)²=1
  • (sina) to the power of 4/(sinx) to the power of 2+(cosa) to the power of 4/(cosx) to the power of 2=1
  • sina^4/sinx^2+cosa^4/cosx^2=1
  • (sina)^4 divide by (sinx)^2+(cosa)^4 divide by (cosx)^2=1

  • Similar expressions

  • (sina)^4/(sinx)^2-(cosa)^4/(cosx)^2=1

(sina)^4/(sinx)^2+(cosa)^4/(cosx)^2=1 equation

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Numerical solution:

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The solution

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   4         4       
sin (a)   cos (a)    
------- + ------- = 1
   2         2       
sin (x)   cos (x)
$$\frac{\sin^{4}{\left(a \right)}}{\sin^{2}{\left(x \right)}} + \frac{\cos^{4}{\left(a \right)}}{\cos^{2}{\left(x \right)}} = 1$$
Simplify
The graph
Rapid solution [src] 
           /    /  1   \\         /    /  1   \\
x1 = - 2*re|atan|------|| - 2*I*im|atan|------||
           |    |   /a\||         |    |   /a\||
           |    |tan|-|||         |    |tan|-|||
           \    \   \2///         \    \   \2///
$$x_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}$$ 
         /    /  1   \\         /    /  1   \\
x2 = 2*re|atan|------|| + 2*I*im|atan|------||
         |    |   /a\||         |    |   /a\||
         |    |tan|-|||         |    |tan|-|||
         \    \   \2///         \    \   \2///
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}$$ 
           /    /   /a\\\         /    /   /a\\\
x3 = - 2*re|atan|tan|-||| - 2*I*im|atan|tan|-|||
           \    \   \2///         \    \   \2///
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}$$ 
         /    /   /a\\\         /    /   /a\\\
x4 = 2*re|atan|tan|-||| + 2*I*im|atan|tan|-|||
         \    \   \2///         \    \   \2///
$$x_{4} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}$$ 
x4 = 2*re(atan(tan(a/2))) + 2*i*im(atan(tan(a/2)))
Sum and product of roots [src] 
sum
      /    /  1   \\         /    /  1   \\       /    /  1   \\         /    /  1   \\         /    /   /a\\\         /    /   /a\\\       /    /   /a\\\         /    /   /a\\\
- 2*re|atan|------|| - 2*I*im|atan|------|| + 2*re|atan|------|| + 2*I*im|atan|------|| + - 2*re|atan|tan|-||| - 2*I*im|atan|tan|-||| + 2*re|atan|tan|-||| + 2*I*im|atan|tan|-|||
      |    |   /a\||         |    |   /a\||       |    |   /a\||         |    |   /a\||         \    \   \2///         \    \   \2///       \    \   \2///         \    \   \2///
      |    |tan|-|||         |    |tan|-|||       |    |tan|-|||         |    |tan|-|||                                                                                          
      \    \   \2///         \    \   \2///       \    \   \2///         \    \   \2///
$$\left(\left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}\right)\right) + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}\right)$$ 
=
0
$$0$$ 
product
/      /    /  1   \\         /    /  1   \\\ /    /    /  1   \\         /    /  1   \\\ /      /    /   /a\\\         /    /   /a\\\\ /    /    /   /a\\\         /    /   /a\\\\
|- 2*re|atan|------|| - 2*I*im|atan|------|||*|2*re|atan|------|| + 2*I*im|atan|------|||*|- 2*re|atan|tan|-||| - 2*I*im|atan|tan|-||||*|2*re|atan|tan|-||| + 2*I*im|atan|tan|-||||
|      |    |   /a\||         |    |   /a\||| |    |    |   /a\||         |    |   /a\||| \      \    \   \2///         \    \   \2//// \    \    \   \2///         \    \   \2////
|      |    |tan|-|||         |    |tan|-|||| |    |    |tan|-|||         |    |tan|-||||                                                                                          
\      \    \   \2///         \    \   \2//// \    \    \   \2///         \    \   \2////
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}\right)$$ 
=
                                          2                                        2
   /    /    /  1   \\     /    /  1   \\\  /    /    /   /a\\\     /    /   /a\\\\ 
16*|I*im|atan|------|| + re|atan|------||| *|I*im|atan|tan|-||| + re|atan|tan|-|||| 
   |    |    |   /a\||     |    |   /a\|||  \    \    \   \2///     \    \   \2//// 
   |    |    |tan|-|||     |    |tan|-||||                                          
   \    \    \   \2///     \    \   \2////
$$16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)}\right)}\right)^{2} \left(\operatorname{re}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\tan{\left(\frac{a}{2} \right)} \right)}\right)}\right)^{2}$$ 
16*(i*im(atan(1/tan(a/2))) + re(atan(1/tan(a/2))))^2*(i*im(atan(tan(a/2))) + re(atan(tan(a/2))))^2
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