SOLUTION: Without using a calculator calculate the value of sec165

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Question 841560: Without using a calculator calculate the value of sec165
Answer by alicealc(293) About Me  (Show Source):
You can put this solution on YOUR website!
All angles below are in degrees

sec(165) = sec(120+45)
=1%2F%28cos%28120%2B45%29%29

(*)Using the formula:
cos(a + b) = cos(a)*cos(b) - sin(a)*sin(b)

=1%2F%28cos%28120%29%2Acos%2845%29-sin%28120%29%2Asin%2845%29%29%29
=1%2F%28cos%28180-60%29%2Acos%2845%29-sin%28180-60%29%2Asin%2845%29%29%29
=1%2F%28-cos%2860%29%2Acos%2845%29-sin%2860%29%2Asin%2845%29%29%29
=
=1%2F%28%28-1%2F4%29%2Asqrt%282%29-%281%2F4%29%2Asqrt%286%29%29%29%29
=1%2F%28%28-sqrt%282%29-sqrt%286%29%29%2F4%29
=4%2F%28%28-sqrt%282%29-sqrt%286%29%29%29
=4%2F%28-%28sqrt%282%29%2Bsqrt%286%29%29%29
=-4%2F%28sqrt%282%29%2Bsqrt%286%29%29
=

(*)Remember:
%28a%2Bb%29%2A%28a-b%29=a%5E2-b%5E2

=-4%28sqrt%282%29-sqrt%286%29%29%2F%282-6%29
=-4%28sqrt%282%29-sqrt%286%29%29%2F-4
=sqrt%282%29-sqrt%286%29

so,
sec%28165%29=sqrt%282%29-sqrt%286%29