Answer:
Use the Sum & Difference Identity: cos (A + B) = cos A · cos B - sin A · sin B
Recall the following from Unit Circle: cos (π/2) = 0, sin (π/2) = 1
cos (π) = -1, sin (π) = 0
Use the Quotient Identity: \tan A=\dfrac{\sin A}{\cos A}tanA=
cosA
sinA
Proof LHS → RHS:
\text{LHS:}\qquad \qquad \dfrac{\cos \bigg(\dfrac{\pi}{2}+x\bigg)}{\cos \bigg(\pi +x\bigg)}LHS:
cos(π+x)
cos(
2
π
+x)
\text{Sum Difference:}\qquad \dfrac{\cos \dfrac{\pi}{2}\cdot \cos x-\sin \dfrac{\pi}{2}\cdot \sin x}{\cos \pi \cdot \cos x-\sin \pi \cdot \sin x}Sum Difference:
cosπ⋅cosx−sinπ⋅sinx
cos
2
π
⋅cosx−sin
2
π
⋅sinx
\text{Unit Circle:}\qquad \qquad \dfrac{0\cdot \cos x-1\cdot \sin x}{-1\cdot \cos x-0\cdot \sin x}Unit Circle:
−1⋅cosx−0⋅sinx
0⋅cosx−1⋅sinx
=\dfrac{-\sin x}{-\cos x}=
−cosx
−sinx
Quotient: tan x
LHS = RHS \checkmark✓
