3.1. Không dùng bảng số hay máy tính cầm tay, tính giá trị của các biểu thức sau:
a) $\left(2 \sin 30^{\circ}+\cos 135^{\circ}-3 \tan 150^{\circ}\right)\left(\cos 180^{\circ}-\cot 60^{\circ}\right)$;
b) $\sin ^2 90^{\circ}+\cos ^2 120^{\circ}+\cos ^2 0^{\circ}-\tan ^2 60^{\circ}+\cot ^2 135^{\circ}$;
c) $\cos 60^{\circ} \cdot \sin 30^{\circ}+\cos ^2 30^{\circ}$.
Chú ý: $\sin ^2 \alpha=(\sin \alpha)^2 ; \cos ^2 \alpha=(\cos \mathrm{x} \alpha)^2 ; \tan ^2 \alpha=(\tan \alpha)^2 ; \cot ^2 \alpha=(\cot \alpha)^2$.
Lời giải.
$$
\text { a) } \begin{aligned}
& \left(2 \sin 30^{\circ}+\cos 135^{\circ}-3 \tan 150^{\circ}\right)\left(\cos 180^{\circ}-\cot 60^{\circ}\right) \\
= & \left(2 \sin 30^{\circ}+\cos \left(180^{\circ}-45^{\circ}\right)-3 \tan \left(180^{\circ}-30^{\circ}\right)\right)\left(\cos 180^{\circ}-\cot 60^{\circ}\right) \\
= & \left(2 \sin 30^{\circ}-\cos 45^{\circ}+3 \tan 30^{\circ}\right)\left(-1-\cot 60^{\circ}\right) \\
= & \left(2 \cdot \frac{1}{2}-\frac{\sqrt{2}}{2}+3 \cdot \frac{1}{\sqrt{3}}\right)\left(-1-\frac{1}{\sqrt{3}}\right) \\
= & -\frac{(2-\sqrt{2}+2 \sqrt{3})(\sqrt{3}+1)}{2 \sqrt{3}} .
\end{aligned}
$$

$$
\begin{aligned}
& \text { b) } \sin ^2 90^{\circ}+\cos ^2 120^{\circ}+\cos ^2 0^{\circ}-\tan ^2 60^{\circ}+\cot ^2 135^{\circ} \\
& =\left(\sin 90^{\circ}\right)^2+\left(\cos 120^{\circ}\right)^2+\left(\cos 0^{\circ}\right)^2-\left(\tan 60^{\circ}\right)^2+\left(\cot 135^{\circ}\right)^2 \\
& =1+\left(\cos \left(180^{\circ}-60^{\circ}\right)\right)^2+1-(\sqrt{3})^2+\left(\cot \left(180^{\circ}-45^{\circ}\right)\right)^2 \\
& =1+\left(\cos 60^{\circ}\right)^2+1-(\sqrt{3})^2+\left(\cot 45^{\circ}\right)^2=\frac{1}{4} .
\end{aligned}
$$
c) $\cos 60^{\circ} \cdot \sin 30^{\circ}+\cos ^2 30^{\circ}=\frac{1}{2} \cdot \frac{1}{2}+\left(\cos 30^{\circ}\right)^2=1$
3.2. Đơn giản biểu thức sau:
a) $\sin 100^{\circ}+\sin 80^{\circ}+\cos 16^{\circ}+\cos 164^{\circ}$.
b) $2 \sin \left(180^{\circ}-\alpha\right) \cdot \cot \alpha+\cos \left(180^{\circ}-\alpha\right) \cdot \tan \alpha \cdot \cot \left(180^{\circ}-\alpha\right)$ với $0^{\circ}<\alpha<90^{\circ}$.
Lời giải.
$$
\text { a) } \begin{aligned}
& \sin 100^{\circ}+\sin 80^{\circ}+\cos 16^{\circ}+\cos 164^{\circ} \\
&= \sin \left(180^{\circ}-80^{\circ}\right)+\sin 80^{\circ}+\cos 16^{\circ}+\cos \left(180^{\circ}-16^{\circ}\right) \\
&= \sin 80^{\circ}+\sin 80^{\circ}+\cos 16^{\circ}-\cos 16^{\circ}=2 \sin 80^{\circ} .
\end{aligned}
$$

b) $2 \sin \left(180^{\circ}-\alpha\right) \cdot \cot \alpha+\cos \left(180^{\circ}-\alpha\right) \cdot \tan \alpha \cdot \cot \left(180^{\circ}-\alpha\right)$

$=2 \sin \alpha \cdot \cot \alpha+\cos \alpha \cdot \tan \alpha \cdot \cot \alpha=2 \sin \alpha \cdot \frac{\cos \alpha}{\sin \alpha}+\cos \alpha=3 \cos \alpha$.

3.4. Cho góc $\alpha\left(0^{\circ}<\alpha<180^{\circ}\right)$ thỏa mãn $\tan \alpha=3$.
Tính giá trị của biểu thức $P=\frac{2 \sin \alpha-3 \cos \alpha}{3 \sin \alpha+2 \cos \alpha}$.
Lời giải.
Ta có $\tan \alpha=3 \Rightarrow \cos \alpha \neq 0$ nên chia cả tử và mẫu của biểu thức $P$ cho $\cos \alpha$ ta được
$$
P=\frac{2 \sin \alpha-3 \cos \alpha}{3 \sin \alpha+2 \cos \alpha}=\frac{2 \tan \alpha-3}{3 \tan \alpha+2}=\frac{3}{11} \text {. }
$$

Câu 2. Tính giá trị các biểu thức sau:
a) $A=\sin ^2 3^{\circ}+\sin ^2 15^{\circ}+\sin ^2 75^{\circ}+\sin ^2 87^{\circ}$
b) $B=\cos 0^{\circ}+\cos 20^{\circ}+\cos 40^{\circ}+\ldots+\cos 160^{\circ}+\cos 180^{\circ}$
c) $C=\tan 5^{\circ} \tan 10^{\circ} \tan 15^{\circ} \ldots \tan 80^{\circ} \tan 85^{\circ}$
Lời giải:
$$
\text { a) } \begin{aligned}
A=\left(\sin ^2 3^{\circ}+\sin ^2 87^{\circ}\right)+\left(\sin ^2 15^{\circ}+\sin ^2 75^{\circ}\right) \\
=\left(\sin ^2 3^{\circ}+\cos ^2 3^{\circ}\right)+\left(\sin ^2 15^{\circ}+\cos ^2 15^{\circ}\right)=1+1=2
\end{aligned}
$$
$$
\begin{aligned}
& \text { b) } B=\left(\cos 0^{\circ}+\cos 180^{\circ}\right)+\left(\cos 20^{\circ}+\cos 160^{\circ}\right)+\ldots+\left(\cos 80^{\circ}+\cos 100^{\circ}\right) \\
& =\left(\cos 0^{\circ}-\cos 0^{\circ}\right)+\left(\cos 20^{\circ}-\cos 20^{\circ}\right)+\ldots+\left(\cos 80^{\circ}-\cos 80^{\circ}\right)=0 \\
&
\end{aligned}
$$

$\begin{aligned} & \text { c) } C=\left(\tan 5^{\circ} \tan 85^{\circ}\right)\left(\tan 15^{\circ} \tan 75^{\circ}\right) \ldots\left(\tan 45^{\circ} \tan 45^{\circ}\right) \\ & =\left(\tan 5^{\circ} \cot 5^{\circ}\right)\left(\tan 15^{\circ} \cot 5^{\circ}\right) \ldots\left(\tan 45^{\circ} \cot 5^{\circ}\right)=1\end{aligned}$