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Answer 1 · 2017-04-23T20:27:26

$2.625$ $color(red)2.625$

$sec 292.5 = sec (\frac{585}{2}) = \frac{1}{cos (\frac{585}{2})}$ $sec 292.5 = sec (585/2) = 1/cos(585/2)$

Let us calculate $cos (\frac{585}{2})$ $cos (585/2)$

$cos (\frac{585}{2}) = cos [\frac{720 - 135}{2}] = cos (360 - \frac{135}{2})$ $cos (585/2) = cos [(720-135)/2] = cos(360 - 135/2)$

$= cos (- \frac{135}{2}) = cos (\frac{135}{2})$ $= cos(-135/2) = cos(135/2)$ . ----------(1.)

Now,
$cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos x}{2}}$ $cos (x/2) = +-sqrt[(1+cosx)/2]$

$therefore cos(135/2) = +-sqrt[(1+cos135)/2]$

$cos 135 = cos (180 - 45) = -cos 45 = -1/sqrt2 = -0.71$

$=> cos(135/2) = +-sqrt[(1-0.71)/2]$

$= +- sqrt(0.29/2) = +-sqrt(.0145) = +-0.381$

but $135/2 = 67.5 < 90.$ Hence $cos(135/2)$ will be positive.

$therefore cos(135/2) = 0.381$

From (1.) $cos (585/2) = cos(135/2) = 0.381$

$=> sec 292.5 = sec (585/2) = 1/cos(585/2) = 1/.381 = color(red)2.625$

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