Quandaries and Queries


My name is Jesus and I´m a secondary student. I´m trying to verify a trigonometric identity but I don´t know how to do it.please help me!!!!!!!!!.

The identity is : tan55º.tan65º.tan75º=tan85º (the sign (.) indicates multiplication)

Thank you.



Hi Jesus,

This is not an identity, it is an equation. A trigonometric identity has variables in it and teh statement is true regardess of the values of the variables.

This result is true but we don't see any nice way to see it. We would like to prove an identity that, for a specific value of the variable, gives your equation, but we don't see how. Anyway you can prove the expression is true using the fillowing identities.

sinA sinB = 1/2[cos(A - B) - cos(A + B)]

cosA cosB = 1/2[cos(A - B) + cos(A + B)]

sinA cosB = 1/2[sin(A - B) + sin(A + B)]


(tan55)(tan65)(tan75) = tan85


sin65/cos65] [ sin75/cos75] = [ sin85/cos85] [ cos55/sin55]

or, using the identities above again

 (cos10 - cos140)/(cos10 + cos140)(sin30 + sin140)/(-sin30 + sin140)

Using cos140 = cos(180 - 40) = -cos40, sin30 = cos60 and similar expressions, the equation to prove can be written as

(cos10 + cos40)/(cos10 - cos40)(cos60 + cos50)/(-cos60 + cos50)

which is

-cos10 cos60 - cos40 cos60 + cos10 cos50 + cos40 cos50
= cos60 cos10 - cos60 cos40 + cos50 cos10 - cos50 cos40

or, finally

cos 10 cos60 = cos40 cos50

Since cos60 = 1/2, the left side is

 1/2 cos10.

Using the second identity at the top of the page one last time, the right side is

cos 40 cos50 = 1/2 (cos10 + cos90) =  1/2 cos10.

Thus your equation is true.

Chris and Penny


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