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More # ${\text{Prove that }}\tan {15^ \circ } + \tan {30^ \circ } + \tan {15^ \circ }\tan {30^ \circ } = 1$ Verified
$\\ {\text{We know that tan4}}{{\text{5}}^ \circ } = 1 \\ {\text{We can write tan4}}{{\text{5}}^ \circ } = \tan ({30^ \circ } + {15^ \circ }) \\ {\text{We also know that tan(}}A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} \\ {\text{By using this we can write }} \\ \Rightarrow \tan ({30^ \circ } + {15^ \circ }) = \dfrac{{\tan {{30}^ \circ } + \tan {{15}^ \circ }}}{{1 - \tan {{30}^ \circ }\tan {{15}^ \circ }}} = \tan {45^ \circ } = 1 \\ {\text{By solving above equation}} \\ \tan {30^ \circ } + \tan {15^ \circ } = 1 - \tan {30^ \circ }\tan {15^ \circ } \\ {\text{By rearranging the equation we get the result}} \\ \tan {30^ \circ } + \tan {15^ \circ } + \tan {30^ \circ }\tan {15^ \circ } = 1{\text{ }}proved \\ {\text{Note: - In such type of question always try to apply the formula of tan(}}A + B){\text{ or tan(}}A - B) \\ {\text{ and put the angles that are given in question so you can prove it}}{\text{.}} \\$