6.1 The Sine Law

\( \frac { \sin ^ { 4 } \alpha + \cos ^ { 4 } \alpha } { \sin ^ { 2 } \alpha \times \cos ^ { 2 } \alpha } - ( \tan \alpha + \cot \alpha ) ^ { 2 } \)

Solve, finding all solutions. Express the solutions in both radians and degrees. \( \tan x=-1 \) Express the solutions in radians. Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. There is one solution on \( [0, \pi) \) at \( x=\frac{3 \pi}{4} \). Solutions are of the form \( x+k \pi \), where \( k \) is an integer. (Simplify your answer. Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) (Simplify your answers. Type exact answers, using \( \pi \) as needed. Use integers or fractions for any numbers in the expressions. Use commas Express the solutions in degrees. Select the correct answer below and, if necessary, fill in the answer box the complete your choice. A. There is one solution on \( \left[0^{\circ}, 180^{\circ}\right. \) ) at \( x=\square^{\circ} \). Solutions are of the form \( x^{\circ}+k \cdot 180^{\circ} \), where \( k \) is an integer. B. There are multiple solutions on \( \left[0^{\circ}, 180^{\circ}\right. \) ) at \( x=\square^{\circ} \). Solutions are of the form \( x^{\circ}+k \cdot 180^{\circ} \), where \( k \) is an integer. (Use commas to separate answers as needed.) C. There are no solutions to the equation.

\( \sin ( 2 x + \frac { \pi } { 6 } ) = \cos x + \cos ( x + \frac { \pi } { 6 } ) \sin x \)

Solve, finding all solutions. Express the solutions in both radians and degrees. \( \sin x=\frac{1}{2} \) Express the solutions in radians. Select the correct answer below and, if necessary, fill in the answer box to complete your choice A. There is one solution on \( [0,2 \pi) \) at \( x=\square \). Solutions are of the form \( x+2 \mathrm{k} \pi \), where k is an integer. (Simplify your answer. Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the express B. There are multiple solutions on \( [0,2 \pi) \) at \( x=\square \). Solutions are of the form \( x+2 \mathrm{k} \pi \), where \( k \) is an integer. (Simplify your answers. Type exact answers, using \( \pi \) as needed. Use integers or fractions for any numbers in the expressio C. There are no solutions to the equation.

Find the exact value of \( \cos 45^{\circ} \)