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Why 1+tan^2 angle=1/cos^2 angle?identity

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rouche | Student, Undergraduate | eNoter

Posted December 19, 2011 at 2:01 AM via web

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Why 1+tan^2 angle=1/cos^2 angle?

identity

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 19, 2011 at 2:07 AM (Answer #1)

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You should come up with the substitution `angle = alpha` .

You need to prove `1 + tan^2 alpha = 1/(cos^2 alpha)`

You should remember that `tan alpha = sin alpha/cos alpha =gt tan^2 alpha = (sin ^2 alpha)/(cos ^2 alpha)`

`` `1 + (sin ^2 alpha)/(cos ^2 alpha) = 1/(cos ^2 alpha)`

Bringing all terms to a common denominator yields;

`(cos ^2 alpha +sin ^2 alpha)/(cos ^2 alpha) = 1/(cos ^2 alpha)`

You need to use the basic formula of trigonometry:

`cos ^2 alpha +sin ^2 alpha = 1`

Using this formula yields:

`1/(cos ^2 alpha) = 1/(cos ^2 alpha)`

Hence, the relation `1 + tan^2 alpha = 1/(cos^2 alpha)`  becomes an identity.

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