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show me the process of solving (1-cot200)(1-cot25)

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show me the process of solving (1-cot200)(1-cot25)

Guest Jul 1, 2015

#1
+19999
+16

show me the process of solving (1-cot200)(1-cot25)

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right]\\\\ &=& \left[ 1 - \dfrac { 1 } { \tan{ ( 200\ensurement{^{\circ}} ) } } \right] \cdot \left[ 1 - \dfrac { 1 } { \tan{ ( 25\ensurement{^{\circ}} ) } } \right]\\\\ &=& \left[ \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } - 1 } { \tan{ ( 200\ensurement{^{\circ}} ) } } \right] \cdot \left[ \dfrac { \tan{ ( 25\ensurement{^{\circ}} ) } - 1 } { \tan{ ( 25\ensurement{^{\circ}} ) } } \right]\\\\ &=& \dfrac { \left[\tan{ ( 200\ensurement{^{\circ}} ) } - 1\right]\cdot \left[ \tan{ ( 25\ensurement{^{\circ}} ) } - 1 \right] } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \tan{ ( 200\ensurement{^{\circ}} ) } - \tan{ ( 25\ensurement{^{\circ}} ) } +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \end{array} }}$$

$$\\\small{ \text{Formula:  \begin{array}{rcl} &&\\ &&\\ &&\\ &&\\ \boxed{ \tan{ (200\ensurement{^{\circ}}+25\ensurement{^{\circ}} ) } = \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } } { 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } } \qquad \tan{ (200\ensurement{^{\circ}}+25\ensurement{^{\circ}} ) } = \tan{ (225\ensurement{^{\circ}} ) } = \tan{ (180\ensurement{^{\circ}}+45\ensurement{^{\circ}} ) } = \tan{ (45\ensurement{^{\circ}} ) } = 1 } \end{array} }} \\\\ \small{ \text{ \begin{array}{rcl} 1 &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } } { 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } &=& \tan{ ( 200\ensurement{^{\circ}} ) + \tan{ ( 25\ensurement{^{\circ}} ) } } \\\\ \end{array} }}$$

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } -1 + \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \end{array} }}$$

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } + \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& 2\cdot \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \mathbf{ \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] }&\mathbf{ =}& \mathbf{2 } \end{array} }}$$

heureka  Jul 2, 2015
#1
+19999
+16

show me the process of solving (1-cot200)(1-cot25)

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right]\\\\ &=& \left[ 1 - \dfrac { 1 } { \tan{ ( 200\ensurement{^{\circ}} ) } } \right] \cdot \left[ 1 - \dfrac { 1 } { \tan{ ( 25\ensurement{^{\circ}} ) } } \right]\\\\ &=& \left[ \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } - 1 } { \tan{ ( 200\ensurement{^{\circ}} ) } } \right] \cdot \left[ \dfrac { \tan{ ( 25\ensurement{^{\circ}} ) } - 1 } { \tan{ ( 25\ensurement{^{\circ}} ) } } \right]\\\\ &=& \dfrac { \left[\tan{ ( 200\ensurement{^{\circ}} ) } - 1\right]\cdot \left[ \tan{ ( 25\ensurement{^{\circ}} ) } - 1 \right] } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \tan{ ( 200\ensurement{^{\circ}} ) } - \tan{ ( 25\ensurement{^{\circ}} ) } +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \end{array} }}$$

$$\\\small{ \text{Formula:  \begin{array}{rcl} &&\\ &&\\ &&\\ &&\\ \boxed{ \tan{ (200\ensurement{^{\circ}}+25\ensurement{^{\circ}} ) } = \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } } { 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } } \qquad \tan{ (200\ensurement{^{\circ}}+25\ensurement{^{\circ}} ) } = \tan{ (225\ensurement{^{\circ}} ) } = \tan{ (180\ensurement{^{\circ}}+45\ensurement{^{\circ}} ) } = \tan{ (45\ensurement{^{\circ}} ) } = 1 } \end{array} }} \\\\ \small{ \text{ \begin{array}{rcl} 1 &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } } { 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } &=& \tan{ ( 200\ensurement{^{\circ}} ) + \tan{ ( 25\ensurement{^{\circ}} ) } } \\\\ \end{array} }}$$

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ \tan{ ( 200\ensurement{^{\circ}} ) } + \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } - \left[ 1 - \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } \right] +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } -1 + \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } +1 } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \end{array} }}$$

$$\small{ \text{ \begin{array}{rcl} \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] &=& \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } + \tan{ ( 200\ensurement{^{\circ}} ) } \cdot \tan{ ( 25\ensurement{^{\circ}} ) } } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ &=& 2\cdot \dfrac { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } } { \tan{ ( 200\ensurement{^{\circ}} ) }\cdot \tan{ ( 25\ensurement{^{\circ}} ) } }\\\\ \mathbf{ \left[ 1 - \cot{(200\ensurement{^{\circ}})} ] \cdot [ 1 - \cot{(25\ensurement{^{\circ}})} \right] }&\mathbf{ =}& \mathbf{2 } \end{array} }}$$

heureka  Jul 2, 2015
#2
+93301
0

That is great Heureka

Melody  Jul 2, 2015