Analytical Trignometry

Question

I have two questions : A)Can someone answer this and explain how he/she answered.

α tan(α+π)=

1. −tanα

2. tanα

3. ctgα

4. None of the above

B) why is cos100 = -cos80

thx in advance

Answer 1

note that $$\tan(\alpha+\pi)=\tan(\alpha)$$ since $\pi$ is the period further we have $$\cos(100^{\circ})=\cos(180^{\circ}-80^{\circ})=-\cos(80^{\circ})$$

Answer 2

For part $b$, by using the identity

$$\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$$ $$\cos(100^\circ)=\cos(180^\circ-80^\circ)=\cos(180^\circ)\cos(80^\circ)+\sin(180^\circ)\sin(80^\circ)$$ $$\implies \cos(100^\circ)=-\cos(80^\circ)$$

Because $\cos(180^\circ)=-1$ and $\sin(180^\circ)=0$