Step 1a

Identify the angle measure: In this case, the angle measure is \$(5pi)/6\$.

Identify the trigonometric functions: The trigonometric functions given are sin and cos.

Recall the values of trigonometric functions for common angles:

For \$(5pi)/6\$, we need to use the reference angle \$pi/6\$, and the fact that sine is negative in the third quadrant while cosine is negative in the second quadrant.

So, sin \$(5pi)/6\$ = - sin \$pi/6\$ = \$-1/2\$
and cos \$(5pi)/6\$ = - cos \$pi/6\$ = \$-\sqrt3/2\$

Therefore, sin \$(5pi)/6\$ = \$- 1/2\$ and

cos \$(5pi)/6\$ = \$-\sqrt3/2\$.

Explanation: Here, identify the angle and trigonometric function, then find \$sin((5π)/6) = -1/2\$ and \$cos((5π)/6) = -\sqrt(3)/2\$.

Step 2a

The given expression is simplify
⇒ cos(x + y) = 1
⇒ \$x + y = cos^-1(1)\$
⇒ x + y = 360°
⇒ y = 360° - x
⇒ x = 360° - y

Explanation: Simplify the expression given, then determine the value of x.

Step 2b

Use the formula:
\$"sin"("x" - "y") = "sinx" "cosy" - "cosx" "siny"\$
\$"sin"("A" - "B") = "sinA" "cosB" - "cosA" "sinB"\$

Then plug the "x" value in the above formula and simplify the expression
⇒ \$"sin"(360° - "y") "cosy" - "cos"(360° - "y") "siny"\$
⇒ \$-"sin"("y") "cos"("y") - "cos"("y") "sin"("y")\$
⇒ \$-2"sin"("y") "cos"("y")\$
⇒ -sin2y

Explanation: Apply the formula, substitute the value of x, and then simplify to obtain the expression as - sin2y.