What is $sectheta*csctheta$ in terms of $costheta$ ?

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Answer 1 · 2016-01-20T11:03:33

Step by step working is shown below.

$color(blue)(sec(theta)*csc(theta)$

We know

$color(green)(sec(theta) =1/cos(theta)$

$color(green)(csc(theta) = 1/sin(theta)$

$color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)$

Note apply the Pythagorean identity

$color(green)(cos^2(theta)+sin^2(theta) = 1$
$color(green)(=> sin^2(theta) = (1-cos^2(theta))$

$color(green)(=>sin(theta) = sqrt(1-cos^2(theta))$

Our problem would become
$sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)$

$color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sqrt(1-cos^2(theta)$

$color(blue)(sec(theta)*csc(theta) = 1/(cos(theta)sqrt(1-cos^2(theta))$

What is sectheta*csctheta sectheta*csctheta in terms of costhetacostheta?