1.

${(\sqrt{a}+\sqrt{b})}^2=a+b$

2.

$\sqrt{a^2-b^2}=a-b$

3.

${\displaystyle \frac{1}{a+b}}={\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}$

4.

${\displaystyle \frac{a+b}{a}}=b$

5.

$\tan (\alpha +\beta )={\displaystyle \frac{\sin (\alpha +\beta )}{\cos (\alpha +\beta )}}$

6.

${\displaystyle \frac{1}{\tan \theta }}={\displaystyle \frac{\cos \theta }{\sin \theta }}$

7.

$(1+\tan \theta {)}^2=1+{\tan }^2\theta$

8.

$\sqrt{1-{\sin }^2\varphi }=1-\sin \varphi$

9.

$\sin 2t=2\sin t$

10.

$\cos \theta +\sin \theta =1$

11.

$\sin (30°+\beta )={\displaystyle \frac{1}{2}}+\sin \beta$

12.

$\cos (90°-C)=\sin C$

13.

$\tan (90°-\theta )={\displaystyle \frac{1}{\tan \theta }}$

14.

$\tan 2\theta ={\displaystyle \frac{2\tan \theta }{1-{\tan }^2\theta }}$

15.

${\displaystyle \frac{{\tan }^{2}x}{1+{\tan }^{2}x}}={\sin }^{2}x$

16.

$\tan x+{\displaystyle \frac{1}{\tan x}}=\sin x\cos x$

17.

$(1+\sin w)(1-\sin w)={\cos }^{2}w$

18.

$(\cos \theta -1)(\cos \theta +1)=-{\sin }^2\theta$

19.

$(\cos \theta -\sin \theta {)}^2=1-2\sin \theta \cos \theta$

20.

${\sin }^{2}x-{\cos }^{2}x=1-2{\cos }^{2}x$

21.

$\tan \theta \cos \theta =\sin \theta$

22.

${\displaystyle \frac{\sin \mu }{\tan \mu }}=\cos \mu$

23.

${\cos }^{4}x-{\sin }^{4}x={\cos }^{2}x-{\sin }^{2}x$

24.

$1-2{\cos }^{2}v+{\cos }^{4}v={\sin }^{4}v$

25.

${\displaystyle \frac{\sin u}{1+\cos u}}={\displaystyle \frac{1-\cos u}{\sin u}}$

Hint.

Multiply numerator and denominator of the left side by $1-\cos u.$

26.

${\displaystyle \frac{\sin v}{1-\cos v}}={\displaystyle \frac{\tan v(1+\sin v)}{\cos v}}$

Hint.

Multiply numerator and denominator of the left side by $1+\sin v.$

27.

${\displaystyle \frac{1}{{\cos }^2\beta }}-{\displaystyle \frac{{\sin }^2\beta }{{\cos }^2\beta }}$

28.

${\displaystyle \frac{1}{{\sin }^2\varphi }}-{\displaystyle \frac{1}{{\tan }^2\varphi }}$

29.

${\cos }^2\alpha (1+{\tan }^2\alpha )$

30.

${\cos }^3\varphi +{\sin }^2\varphi \cos \varphi$

31.

${\tan }^{2}A-{\tan }^{2}A{\sin }^{2}A$

32.

${\cos }^{2}B{\tan }^{2}B+{\cos }^{2}B$

33.

${\displaystyle \frac{1-{\cos }^{2}z}{{\cos }^{2}z}}$

34.

${\displaystyle \frac{\sin t}{\cos t\tan t}}$

35.

$3{\cos }^{2}1.7°+3{\sin }^{2}1.7°$

36.

$4-{\cos }^{2}338°-{\sin }^{2}338°$

37.

$({\cos }^{2}20°+{\sin }^{2}20°{)}^4$

38.

${\displaystyle \frac{18}{{\cos }^{2}17°+{\sin }^{2}17°}}$

39.

${\displaystyle \frac{6}{{\cos }^{2}53°}}-6{\tan }^{2}53°$

40.

${\displaystyle \frac{1}{{\sin }^{2}102°}}-{\displaystyle \frac{{\cos }^{2}102°}{{\sin }^{2}102°}}$

41.

$2{\cos }^2\theta -1=?$

42.

$1-2{\sin }^2\left({\displaystyle \frac{\theta }{2}}\right)=?$

43.

$1-{\displaystyle \frac{{\sin }^{2}x}{1+\cos x}}=?$

44.

${\displaystyle \frac{\sin x}{\sqrt{1-{\sin }^{2}x}}}=?$

45.

$2\tan t{\cos }^{2}t=?$

46.

${\displaystyle \frac{2\tan t}{1-{\tan }^{2}t}}=?$

47.

$2-{\cos }^2\theta +2\sin \theta$ as an expression in $\sin \theta$ only

48.

$3{\sin }^{2}B+2\cos B-4$ as an expression in $\cos B$ only

49.

${\cos }^2\varphi -2{\sin }^2\varphi$ as an expression in $\cos \varphi$ only

50.

${\cos }^2\varphi {\sin }^2\varphi$ as an expression in $\sin \varphi$ only

51.

$\cos \theta -{\sin }^2\theta +1=0$

52.

$4\sin \theta +2{\cos }^2\theta -3=-1$

53.

$1-\sin \theta -2{\cos }^2\theta =0$

54.

$3{\cos }^2\theta -{\sin }^2\theta =2$

55.

$2\cos \theta \tan \theta +1=0$

56.

$\cos \theta -\sin \theta =0$

57.

${\displaystyle \frac{1}{3}}\cos \theta =\sin \theta$

58.

$5\sin C=2\cos C$

59.

$\cos A={\displaystyle \frac{12}{13}}$ and $270°<A<360°$

60.

$\sin B={\displaystyle \frac{-3}{5}}$ and $180°<B<270°$

61.

$\sin \varphi ={\displaystyle \frac{1}{7}}$ and $90°<\varphi <180°$

62.

$\cos t={\displaystyle \frac{-2}{3}}$ and $180°<t<270°$

63.

$\tan \theta =-{\displaystyle \frac{1}{2}}$ and $270°<\theta <360°$

64.

$\tan \theta =2$ and $180°<\theta <270°$

65.

$\tan \theta ={\displaystyle \frac{3}{4}}$ and $180°<\theta <270°$

66.

$\tan \theta =-3$ and $90°<\theta <180°$

67.

$2\cos A+9=8$ and $90°<A<180°$

68.

$25\sin B+8=-12$ and $180°<B<270°$

69.

$8\tan \beta +5=-11$ and $90°<\beta <180°$

70.

$6(\tan \beta -4)=-24$ and $90°<\beta <270°$

71.

${\tan }^{2}C-{\displaystyle \frac{1}{4}}=0$ and $0°<C<180°$

72.

$4{\cos }^{2}A-\cos A=0$ and $00°<A<180°$

73.

${\displaystyle \frac{\tan \alpha }{1+\tan \alpha }}={\displaystyle \frac{\sin \alpha }{\sin \alpha +\cos \alpha }}$

74.

${\displaystyle \frac{1-\tan u}{1+\tan u}}={\displaystyle \frac{\cos u-\sin u}{\cos u+\sin u}}$

75.

${\displaystyle \frac{1+{\tan }^2\beta }{1-{\tan }^2\beta }}={\displaystyle \frac{1}{{\cos }^2\beta -{\sin }^2\beta }}$

76.

${\tan }^{2}v-{\sin }^{2}v={\tan }^{2}v{\sin }^{2}v$

77.

Prove the Pythagorean identity ${\cos }^2\theta +{\sin }^2\theta =1$ by carrying out the following steps. Sketch an angle $\theta$ in standard position and label a point $(x,y)$ on the terminal side, at a distance $r$ from the vertex.

1. Begin with the equation $\sqrt{x^2+y^2}=r,$ and square both sides.
2. Divide both sides of your equation from part (a) by $r^2.$
3. Write the left side of the equation as the sum of the squares of two fractions.
4. Substitute the appropriate trigonometric ratio for each fraction.

78.

Prove the tangent identity $\tan \theta ={\displaystyle \frac{\sin \theta }{\cos \theta }}$ by carrying out the following steps. Sketch an angle $\theta$ in standard position and label a point $(x,y)$ on the terminal side, at a distance $r$ from the vertex.

1. Write $\sin \theta$ in terms of $y$ and $r,$ and solve for $y.$
2. Write $\cos \theta$ in terms of $x$ and $r,$ and solve for $x.$
3. Write $\tan \theta$ in terms of $x$ and $y,$ then substitute your results from parts (a) and (b).
4. Simplify your fraction in part (c).