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How do you simplify $3(tan^2 theta - sec^2 theta)$ ?

2 Answers

Aug 5, 2015

$3(tan^2(theta)-sec^2(theta))=- 3$

Explanation:

$tan(theta) = (sin(theta))/(cos(theta))$
$color(white)("XXXX")$ $color(white)("XXXX")$ $rarr tan^2(theta) = sin^2(theta)/cos^2(theta)$

$sec(theta) = 1/cos(theta)$
$color(white)("XXXX")$ $color(white)("XXXX")$ $sec^2(theta)=1/cos^2(theta)$

$tan^2(theta) - sec^2(theta)$
$color(white)("XXXX")$ $color(white)("XXXX")$ $=sin^2(theta)/cos^2(theta) - 1/cos^2(theta)$

$color(white)("XXXX")$ $color(white)("XXXX")$ $=(sin^2(theta)-1)/(cos^2(theta)$

and since $sin^2(theta)+cos^2(theta) = 1$
$color(white)("XXXX")$ $color(white)("XXXX")$ $= (-cos^2(theta))/cos^2(theta)$

$color(white)("XXXX")$ $color(white)("XXXX")$ $= - 1$

$3(tan^2(theta)-sec^2(theta))$
$color(white)("XXXX")$ $color(white)("XXXX")$ $=- 3$

Aug 6, 2015

You can use the identity:

$1+tan^2theta = sec^2theta$

$=> 3(tan^2theta - (1+tan^2theta))$

$= 3(tan^2theta - 1 - tan^2theta)$

$= 3(-1) = color(blue)(-3)$

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