find the value of $\frac{\tan35⁰tan70⁰tan75⁰}{\tan35⁰+tan70⁰+tan75⁰}$

asked by guest
on Sep 20, 2024 at 12:33 am



You asked:

Evaluate the expression: \(\frac{\tan\left( {35}^{0} \right) \cdot \tan\left( {70}^{0} \right) \cdot \tan\left( {75}^{0} \right)}{\tan\left( {35}^{0} \right) + \tan\left( {70}^{0} \right) + \tan\left( {75}^{0} \right)}\)

MathBot Answer:

\[\frac{\tan\left( {35}^{0} \right) \cdot \tan\left( {70}^{0} \right) \cdot \tan\left( {75}^{0} \right)}{\tan\left( {35}^{0} \right) + \tan\left( {70}^{0} \right) + \tan\left( {75}^{0} \right)} = \frac{\tan^{2}{\left(1 \right)}}{3} \approx 0.80850627360491992031389297784705\]

\[\begin{aligned}\frac{\tan\left( {35}^{0} \right) \cdot \tan\left( {70}^{0} \right) \cdot \tan\left( {75}^{0} \right)}{\tan\left( {35}^{0} \right) + \tan\left( {70}^{0} \right) + \tan\left( {75}^{0} \right)}& = \tan{\left(35^{0} \right)} \cdot \tan{\left(70^{0} \right)} \cdot \tan{\left(75^{0} \right)} \cdot \frac{1}{\tan{\left(35^{0} \right)} + \tan{\left(70^{0} \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(70^{0} \right)} \cdot \tan{\left(75^{0} \right)} \cdot \frac{1}{\tan{\left(35^{0} \right)} + \tan{\left(70^{0} \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(75^{0} \right)} \cdot \frac{1}{\tan{\left(35^{0} \right)} + \tan{\left(70^{0} \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{\tan{\left(35^{0} \right)} + \tan{\left(70^{0} \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{\tan{\left(1 \right)} + \tan{\left(70^{0} \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{\tan{\left(1 \right)} + \tan{\left(1 \right)} + \tan{\left(75^{0} \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{\tan{\left(1 \right)} + \tan{\left(1 \right)} + \tan{\left(1 \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{2 \cdot \tan{\left(1 \right)} + \tan{\left(1 \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{3 \cdot \tan{\left(1 \right)}}\\& = \tan{\left(1 \right)} \cdot \tan{\left(1 \right)} \cdot \frac{1}{3}\\& = \tan^{2}{\left(1 \right)} \cdot \frac{1}{3}\\& = \frac{1}{3} \cdot \tan^{2}{\left(1 \right)}\end{aligned}\]

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