Prove that`(tan 25^@-tan 35^@ +tan 85^@)/(tan 25^@ tan 35^@ tan 85^@)`= 3

Prove that`(tan 25^@-tan 35^@ +tan 85^@)/(tan 25^@ tan 35^@ tan 85^@)`

1.	Prove that`(tan 25^@-tan 35^@ +tan 85^@)/(tan 25^@ tan 35^@ tan 85^@)`= 3
Answer» Here, we will use, `tanxtan(60^@-x)tan(60^@+x) = tan3x` `tanx+tan(60^@+x)+tan(120^@+x) = 3tan3x` Now, `L.H.S. = (tan25^@ - tan35^@+tan85^@)/(tan25^@ tan35^@tan85^@)` `= (tan25^@ + tan(180-35)^@+tan85^@)/(tan25^@ tan(60-25)^@tan(60+25)^@)` `= (tan25^@ +tan85^@+tan 145^@)/(tan25^@ tan(60-25)^@tan(60+25)^@)` `= (tan25^@ +tan(60+25)^@+tan(120+25)^@)/(tan25^@ tan(60-25)^@tan(60+25)^@)` Now, using the above identities, `=(3tan75^@)/tan75^@` `=3 = R.H.S.`

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Prove that`(tan 25^@-tan 35^@ +tan 85^@)/(tan 25^@ tan 35^@ tan 85^@)`= 3

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