document.write( "Question 678945: find the exact value, if defined: cos((arcsec(-2)+arctan1) \n" ); document.write( "
Algebra.Com's Answer #421771 by jsmallt9(3758)\"\" \"About 
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cos(arccsc(-2) + arctan(1))
\n" ); document.write( "Since the inverse functions return angles, this expression is the cos of the sum of two angles. We can use the cos(A+B) formula:
\n" ); document.write( "COS(A+B) = cos(A)cos(B) - sin(A)sin(B)
\n" ); document.write( "With the \"A\" being arccsc(-2) and the \"B\" being arctan(1) we get:
\n" ); document.write( "cos(arccsc(-2))cos(arctan(1)) - sin(arccsc(-2))sin(arctan(1))

\n" ); document.write( "arccsc(-2) is an angle whose csc is -2. Since sin and csc are reciprocals of each other, an angle whose csc is -2 will have a sin that is -1/2. If we didn't recognize that arccsc(-2) was a special angle, we should definitely recognize that an angle whose sin is -1/2 is a special angle. The reference angle will be 30 degrees (or \"pi%2F6\" radians). Arccsc only returns angles between -90 and 90. The only angle in this range which would have a sin of -1/2 is -30. So now our expression is:
\n" ); document.write( "cos(-30)cos(arctan(1)) - sin(-30)sin(arctan(1))

\n" ); document.write( "arctan(1) is an angle whose tan is 1. We should recognize that 1 is a special angle value for tan. We should know that the reference angle that has a tan of 1 is 45 degrees (or \"pi%2F4\" radians). Arctan, like arccsc, returns angles between -90 and 90. The only angle in this range that has a tan of positive 1 is 45. So now our expression is:
\n" ); document.write( "cos(-30)cos(45) - sin(-30)sin(45)

\n" ); document.write( "Now our expression consists of cos's and sin's of special angles. we should now what these are:
\n" ); document.write( "\"%28sqrt%283%29%2F2%29%28sqrt%282%29%2F2%29+-+%28-1%2F2%29%28sqrt%282%29%2F2%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"sqrt%286%29%2F4+%2Bsqrt%282%29%2F4\"
\n" ); document.write( "This is an exact expression for cos(arccsc(-2) + arctan(1)). It may be acceptable as your answer. Or, since the fractions have the same denominators, we could add them:
\n" ); document.write( "\"%28sqrt%286%29%2Bsqrt%282%29%29%2F4\" \n" ); document.write( "
\n" );