Question 180774
 



{{{sec((17pi)/12)}}} Start with the given trig expression.



{{{1/(cos((17pi)/12))}}} Use the identity {{{sec(A)=1/cos(A)}}}



{{{1/(cos((7pi)/6+pi/4))}}} Expand. Note: {{{(17pi)/12=(7pi)/6+pi/4}}} (convert to degrees if that makes it easier to see)



{{{1/(cos((7pi)/6)cos(pi/4)-sin((7pi)/6)sin(pi/4))}}} Use the identity {{{cos(A+B)=cos(A)cos(B)-sin(A)sin(B)}}} to expand




{{{1/((-sqrt(3)/2)cos(pi/4)-sin((7pi)/6)sin(pi/4))}}} Evaluate the cosine of {{{(7pi)/6}}} to get {{{-sqrt(3)/2}}} (use the unit circle)



{{{1/((-sqrt(3)/2)(sqrt(2)/2)-sin((7pi)/6)sin(pi/4))}}} Evaluate the cosine of {{{pi/4}}} to get {{{sqrt(2)/2}}} (use the unit circle)



{{{1/((-sqrt(3)/2)(sqrt(2)/2)-(-1/2)sin(pi/4))}}} Evaluate the sine of {{{(7pi)/6}}} to get {{{-1/2}}} (use the unit circle)




{{{1/((-sqrt(3)/2)(sqrt(2)/2)-(-1/2)(sqrt(2)/2))}}} Evaluate the sine of {{{pi/4}}} to get {{{sqrt(2)/2}}} (use the unit circle)



{{{1/(-sqrt(6)/4+sqrt(2)/4)}}} Multiply



{{{1/((-sqrt(6)+sqrt(2))/4)}}} Combine the fractions.



{{{1/((sqrt(2)-sqrt(6))/4)}}} Rearrange the terms.



{{{(1/1)(4/(sqrt(2)-sqrt(6)))}}} Multiply the first fraction (which is really {{{1/1}}}) by the reciprocal of the second fraction



{{{4/(sqrt(2)-sqrt(6))}}} Multiply




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Answer:



So {{{sec((17pi)/12)=4/(sqrt(2)-sqrt(6))}}}