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

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on Dec 27, 2024 at 5:06 am



You asked:

Evaluate the expression: 4tan⁡(72)tan⁡(72)+4(1cos⁡(72)−1)(1cos⁡(72)−1)\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:

4tan⁡(72)tan⁡(72)+4(1cos⁡(72)−1)(1cos⁡(72)−1)=4tan⁡2(72)+4(1cos⁡(72)−1)2≈4.10143499716244341931898785909903\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

4tan⁡(72)tan⁡(72)+4(1cos⁡(72)−1)(1cos⁡(72)−1)=4⋅tan⁡(72)⋅tan⁡(72)+4⋅(1cos⁡(72)−1)⋅(1cos⁡(72)−1)=4⋅tan⁡2(72)+4⋅(1cos⁡(72)−1)⋅(1cos⁡(72)−1)=4⋅tan⁡2(72)+4⋅(−1+1cos⁡(72))⋅(1cos⁡(72)−1)=4⋅tan⁡2(72)+4⋅(−1+1cos⁡(72))⋅(−1+1cos⁡(72))=4⋅tan⁡2(72)+(−4+4⋅1cos⁡(72))⋅(−1+1cos⁡(72))=4⋅tan⁡2(72)+(−1+1cos⁡(72))⋅(−4+4⋅1cos⁡(72))\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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