$\sqrt{\4*tan(72)*tan(72)+4(1/cos(72)-1)*(1/cos(72)-1)}$

asked by guest
on Dec 27, 2024 at 5:06 am



You asked:

Evaluate the expression: \(\sqrt{4 \tan\left( 72 \right) \tan\left( 72 \right) + 4 \left(\frac{1}{\cos\left( 72 \right)} - 1\right) \left(\frac{1}{\cos\left( 72 \right)} - 1\right)}\)

MathBot Answer:

\[\sqrt{4 \tan\left( 72 \right) \tan\left( 72 \right) + 4 \left(\frac{1}{\cos\left( 72 \right)} - 1\right) \left(\frac{1}{\cos\left( 72 \right)} - 1\right)} = \sqrt{4 \tan^{2}{\left(72 \right)} + 4 \left(\frac{1}{\cos{\left(72 \right)}} - 1\right)^{2}} \approx 4.10143499716244341931898785909903\]

\[\begin{aligned}\sqrt{4 \tan\left( 72 \right) \tan\left( 72 \right) + 4 \left(\frac{1}{\cos\left( 72 \right)} - 1\right) \left(\frac{1}{\cos\left( 72 \right)} - 1\right)}& = \sqrt{4 \cdot \tan{\left(72 \right)} \cdot \tan{\left(72 \right)} + 4 \cdot \left(\frac{1}{\cos{\left(72 \right)}} - 1\right) \cdot \left(\frac{1}{\cos{\left(72 \right)}} - 1\right)}\\& = \sqrt{4 \cdot \tan^{2}{\left(72 \right)} + 4 \cdot \left(\frac{1}{\cos{\left(72 \right)}} - 1\right) \cdot \left(\frac{1}{\cos{\left(72 \right)}} - 1\right)}\\& = \sqrt{4 \cdot \tan^{2}{\left(72 \right)} + 4 \cdot \left(-1 + \frac{1}{\cos{\left(72 \right)}}\right) \cdot \left(\frac{1}{\cos{\left(72 \right)}} - 1\right)}\\& = \sqrt{4 \cdot \tan^{2}{\left(72 \right)} + 4 \cdot \left(-1 + \frac{1}{\cos{\left(72 \right)}}\right) \cdot \left(-1 + \frac{1}{\cos{\left(72 \right)}}\right)}\\& = \sqrt{4 \cdot \tan^{2}{\left(72 \right)} + \left(-4 + 4 \cdot \frac{1}{\cos{\left(72 \right)}}\right) \cdot \left(-1 + \frac{1}{\cos{\left(72 \right)}}\right)}\\& = \sqrt{4 \cdot \tan^{2}{\left(72 \right)} + \left(-1 + \frac{1}{\cos{\left(72 \right)}}\right) \cdot \left(-4 + 4 \cdot \frac{1}{\cos{\left(72 \right)}}\right)}\end{aligned}\]

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