Basic Trigonometric Identities

Sol A $\sin^2\theta=1-\cos^2\theta=1-(12/13)^2=1-144/169=25/169$, so $\sin\theta=-5/13$ since $\sin\theta<0$ if $\theta$ is in Quadrant IV.

Sol B $\sec^2\theta=\tan^2\theta+1=(3/4)^2+1=9/16+1=25/16$, so $\sec\theta=-5/4$ since $\sec\theta<0$ if $\theta$ is in Quadrant III.

Sol C $\tan^2\theta=\sec^2\theta-1=(25/7)^2-1=625/49-1=576/49$, so $\tan\theta=-24/7$ since $\tan\theta<0$ if $\theta$ is in Quadrant IV. Therefore $\cot\theta=1/(\tan\theta)=1/(-24/7)=-7/24$.

Sol 1 $\sec^2\theta=1+\tan^2\theta=1+9=10$, so $\sec\theta=\sqrt{10}$ since $\sec\theta>0$ for $3\pi/2<\theta<2\pi$. Therefore $\cos\theta=\frac{1}{\sec\theta}=1/\sqrt{10}$.

Sol 2 $\sec^2\theta=1+\tan^2\theta=1+(4/3)^2=1+16/9=25/9$, so $\sec\theta=-5/3$ since $\sec\theta<0$ if $\pi<\theta<3\pi/2$. Then $\cos\theta=\frac{1}{\sec\theta}=-3/5$, so $\sin\theta=(\tan\theta)(\cos\theta)=(4/3)(-3/5)=-4/5$.

Sol 3 Since $\cos\theta=-3/5$, $\sec\theta=\frac{1}{\cos\theta}=-5/3$ and so $\tan^2\theta=\sec^2\theta-1=(-5/3)^2-1=25/9-1=16/9$. Therefore $\tan\theta=4/3$ since $\tan\theta>0$ for $\pi<\theta<3\pi/2$.

Sol 4 $(\sin\theta+\cos\theta)^2$ $=\sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta$ $=(\sin^2\theta+\cos^2\theta)+2\sin\theta\cos\theta$ $=1+2\sin\theta\cos\theta$, which can also be written as $1+\sin 2\theta$.

Sol 5 $(\sec 4\theta-1)(\sec 4\theta+1)=\sec^2 4\theta-1$ $=\tan^2 4\theta$.

Sol 6 $(\csc\theta-\cot\theta)(\csc\theta+\cot\theta)$ $=\csc^2 \theta-\cot^2 \theta=1$.

Sol 7 $\sqrt{9-x^2}=\sqrt{9-(3\sin\theta)^2}=\sqrt{9-9\sin^2\theta}$ $=\sqrt{9(1-\sin^2\theta)}=\sqrt{9\cos^2\theta}=3\cos\theta$ since $\cos\theta\ge 0$ for $-\pi/2\le\theta\le\pi/2$.

Sol 8 $\sqrt{x^2+4}=\sqrt{(2\tan\theta)^2+4}=\sqrt{4\tan^2 \theta+4}$ $=\sqrt{4(\tan^2 \theta+1)}=\sqrt{4\sec^2\theta}=2\sec\theta$ since $\sec\theta>0$ for $-\pi/2<\theta<\pi/2$.

Sol 9 $\frac{\cos^2\theta}{\sin\theta}=\frac{1-\sin^2\theta}{\sin\theta}$ $=\frac{1}{\sin\theta}-\frac{\sin^2\theta}{\sin\theta}=\csc\theta-\sin\theta$.

Sol 10 $\sin^4\theta\cos^3\theta=\sin^4\theta(\cos^2\theta)\cos\theta$ $=\sin^4\theta(1-\sin^2\theta)\cos\theta=(\sin^4\theta-\sin^6\theta)\cos\theta$.

Sol 11 $\sin^5\theta=(\sin^4\theta)\sin\theta=(\sin^2\theta)^2\sin\theta$ $=(1-\cos^2\theta)^2\sin\theta=(1-2\cos^2\theta+\cos^4\theta)\sin\theta$

Sol 12 $\sec^6\theta=(\sec^4\theta)\sec^2\theta=(\sec^2\theta)^2\sec^2\theta$ $=(1+\tan^2\theta)^2\sec^2\theta=(1+2\tan^2\theta+\tan^4\theta)\sec^2\theta$.

Sol 13 $\tan\theta+\cot\theta= \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\the... ...theta\sin\theta}=\frac{1}{\cos\theta}\frac{1}{\sin\theta}= \sec\theta\csc\theta$.

Sol 14

$$\frac{\sec^3\theta}{\tan\theta}= \frac{\sec\theta(\sec^2\thet... ...tan\theta+\frac{1}{\sin\theta}= \sec\theta\tan\theta+\csc\theta$$