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Answer 1 · 2017-04-23T20:27:26

$color(red)2.625$

$sec 292.5 = sec (585/2) = 1/cos(585/2)$

Let us calculate $cos (585/2)$

$cos (585/2) = cos [(720-135)/2] = cos(360 - 135/2)$

$= cos(-135/2) = cos(135/2)$ . ----------(1.)

Now,
$cos (x/2) = +-sqrt[(1+cosx)/2]$

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

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

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

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

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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