How do you express $4 cos theta - sec theta + 2 cot theta$ in terms of $sin theta$ ?

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Answer 1 · 2016-03-18T18:00:36

$[4(1-sin^2theta)sintheta-sintheta+2(1-sin^2theta)]/(sinthetasqrt(1-sin^2theta))$

We shall use the following identities:

$sectheta= 1/costheta$

$cottheta= costheta/sintheta$

Also, $cos^2theta+sin^2theta=1$

$=>cos^2theta=1-sin^2theta color(red)rarr costheta=+-sqrt(1-sin^2theta)$

The $+$ or $-$ depends on the quadrant in which the angle $theta$
is found.

Assuming $theta$ acute we can simply ignore this.

Hence,
$costheta= sqrt(1-sin^2theta)$

$=>1/costheta=1/(sqrt(1-sin^2theta))$

$costheta/sintheta=sqrt(1-sin^2theta)/sintheta$

$=>4costheta-sectheta+2cottheta color(red)rarr 4*(sqrt(1-sin^2theta))-(1/(sqrt(1-sin^2theta)))+2*(sqrt(1-sin^2theta)/sintheta)$

$color(red)rarr [4(1-sin^2theta)sintheta-sintheta+2(1-sin^2theta)]/(sinthetasqrt(1-sin^2theta))$

How do you express 4 cos theta - sec theta + 2 cot theta 4 cos theta - sec theta + 2 cot theta in terms of sin theta sin theta ?