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identity1-tan^2angle/1-cot^angle=1-sec^2angle

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home1h | Student, Undergraduate | (Level 2) eNoter

Posted December 20, 2011 at 1:41 AM via web

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identity

1-tan^2angle/1-cot^angle=1-sec^2angle

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 20, 2011 at 2:01 AM (Answer #1)

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Use the following notation `alpha` =angle.

You need to prove the identity:

`(1-tan^2alpha)/(1-cot^2alpha)=1-sec^2alpha`

You need to remember that tan alpha = sin alpha/cos alpha and cot alpha = cos alpha/sin alpha

`(1-sin^2 alpha)/(cos^2 alpha)/(1-cos^2 alpha)/(sin^2 alpha)=1-sec^2alpha`

The left side seems more complicated, hence you will expand such that:

`((cos^2 alpha-sin^2 alpha)/(cos^2 alpha))/((sin^2 alpha-cos^2 alpha)/(sin^2 alpha))`

Reducing by `cos^2 alpha-sin^2 alpha`  yields:

`(1/(cos^2 alpha))/((-1)/(sin^2 alpha)) = - tan^2 alpha`

There is not much that you may do to the left side, hence you may move to the right side and expand such that:

`1-sec^2alpha = 1 - (1 + tan^2 alpha)`

Opening the brackets yields:

`1-sec^2alpha = - tan^2 alpha`

Since the both sides of the last equation are equal, hence the original identity is checked.

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