About Lesson
(1) Prove the following identities
证明下列恒等式
(a) ![Rendered by QuickLaTeX.com \frac{cos{2}x}{sin{x}-cos{x}}=-cos{x}-sin{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-378a82fcdf6ac171fdcc0733f34e0228_l3.png)
![Rendered by QuickLaTeX.com \frac{cos{2}x}{sin{x}-cos{x}}=-cos{x}-sin{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-378a82fcdf6ac171fdcc0733f34e0228_l3.png)
(b) ![Rendered by QuickLaTeX.com \frac{(sin{\frac{x}{2}}+cos{\frac{x}{2}})^2}{1+sin{x}}=1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6486dc90207d3b2cf5265a811fd61080_l3.png)
![Rendered by QuickLaTeX.com \frac{(sin{\frac{x}{2}}+cos{\frac{x}{2}})^2}{1+sin{x}}=1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6486dc90207d3b2cf5265a811fd61080_l3.png)
(c) ![Rendered by QuickLaTeX.com \frac{tan{2}x}{{tan}^2{2}x-1}=-\frac{1}{2}tan{4}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-b41b718b4fae59d9b1496f19d6664d5b_l3.png)
![Rendered by QuickLaTeX.com \frac{tan{2}x}{{tan}^2{2}x-1}=-\frac{1}{2}tan{4}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-b41b718b4fae59d9b1496f19d6664d5b_l3.png)
(d) ![Rendered by QuickLaTeX.com \frac{sin{3}x}{sin{x}}+\frac{cos{3}x}{cos{x}}=4cos{2}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-0b8179ee44398732713bbd95843e68dc_l3.png)
![Rendered by QuickLaTeX.com \frac{sin{3}x}{sin{x}}+\frac{cos{3}x}{cos{x}}=4cos{2}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-0b8179ee44398732713bbd95843e68dc_l3.png)
(e) ![Rendered by QuickLaTeX.com \frac{sin{2}x}{1+cos{2}x}\cdot\frac{cos{x}}{1+cos{x}}=tan{\frac{x}{2}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-14dfd51fe409700623513933e513c88c_l3.png)
![Rendered by QuickLaTeX.com \frac{sin{2}x}{1+cos{2}x}\cdot\frac{cos{x}}{1+cos{x}}=tan{\frac{x}{2}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-14dfd51fe409700623513933e513c88c_l3.png)
(f) ![Rendered by QuickLaTeX.com \frac{sin{2}x+cos{2}x+1}{sin{2}x-cos{2}x+1}=cot{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c3c2d0e0c3f7eb9624579f5d94c39b0c_l3.png)
![Rendered by QuickLaTeX.com \frac{sin{2}x+cos{2}x+1}{sin{2}x-cos{2}x+1}=cot{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c3c2d0e0c3f7eb9624579f5d94c39b0c_l3.png)
(g) ![Rendered by QuickLaTeX.com \frac{1+sin{x}}{1-sin{x}}={tan}^2{(}\frac{\pi}{4}+\frac{x}{2})](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-405537a1fffdafae873b2e1d8375de25_l3.png)
![Rendered by QuickLaTeX.com \frac{1+sin{x}}{1-sin{x}}={tan}^2{(}\frac{\pi}{4}+\frac{x}{2})](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-405537a1fffdafae873b2e1d8375de25_l3.png)
(h) ![Rendered by QuickLaTeX.com {cos}^6{x}+{sin}^6{x}=1-\frac{3}{4}{sin}^2{2}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-93e197570423dc227b8855cddd5ad212_l3.png)
![Rendered by QuickLaTeX.com {cos}^6{x}+{sin}^6{x}=1-\frac{3}{4}{sin}^2{2}x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-93e197570423dc227b8855cddd5ad212_l3.png)
(i) ![Rendered by QuickLaTeX.com \frac{sin{2}A+cos{A}}{cos{2}A+sin{A}}=\frac{cos{A}}{1-sin{A}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d94f5f490e791c79ed10fe425d399723_l3.png)
![Rendered by QuickLaTeX.com \frac{sin{2}A+cos{A}}{cos{2}A+sin{A}}=\frac{cos{A}}{1-sin{A}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d94f5f490e791c79ed10fe425d399723_l3.png)
(j) ![Rendered by QuickLaTeX.com \frac{sin{2}x}{(sin{x}+cos{x}-1)(sin{x}-cos{x}+1)}=cot{\frac{x}{2}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-a27a7e839629508f167e1fd26c911ba3_l3.png)
![Rendered by QuickLaTeX.com \frac{sin{2}x}{(sin{x}+cos{x}-1)(sin{x}-cos{x}+1)}=cot{\frac{x}{2}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-a27a7e839629508f167e1fd26c911ba3_l3.png)
(2)
If ,
and
; find without using calculator or tables, the values of
若,
和
; 不用计算机或查表,求
(a)
(b)
(c)
Answer 答案:
(a) ![Rendered by QuickLaTeX.com -2](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOCIgaGVpZ2h0PSIxNiIgdmlld0JveD0iMCAwIDI4IDE2Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjZWFlYWVjIi8+PC9zdmc+)
(b)![Rendered by QuickLaTeX.com \pm0.811](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2OSIgaGVpZ2h0PSIxNiIgdmlld0JveD0iMCAwIDY5IDE2Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjZWFlYWVjIi8+PC9zdmc+)
(c)![Rendered by QuickLaTeX.com -\frac{4}{5}](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDI5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjZWFlYWVjIi8+PC9zdmc+)
![Rendered by QuickLaTeX.com -2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4363e6f4a04685861bf48830cd2165ce_l3.png)
(b)
![Rendered by QuickLaTeX.com \pm0.811](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-edbb4ca5e8f219454b5a816b48c2d162_l3.png)
(c)
![Rendered by QuickLaTeX.com -\frac{4}{5}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-faa4a1cba1ca865c5cb7dd299edf946f_l3.png)
(3)
Given that
, find the values of a, b and c.
已知
, 求 a, b 和 c的值.
Given that
![Rendered by QuickLaTeX.com 2sin^{2}x + 4sin x cos x + 5cos^{2}x \equiv a + b sin2x + c cos2x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-ba21050c3355d173635ba9f081b3755e_l3.png)
已知
![Rendered by QuickLaTeX.com 2sin^{2}x + 4sin x cos x + 5cos^{2}x \equiv a + b sin2x + c cos2x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-ba21050c3355d173635ba9f081b3755e_l3.png)
Answer 答案:
,
,