Tangent and Cotangent Identities $ \begin{aligned} \tan (\theta) & =\frac{\sin (\theta)}{\cos (\theta)} & \cot (\theta) & =\frac{\cos (\theta)}{\sin (\theta)} \end{aligned}$ $ \begin{aligned} \csc (\theta) & =\frac{1}{\sin (\theta)} & \sin (\theta) & =\frac{1}{\csc (\theta)} \\ \sec (\theta) & =\frac{1}{\cos (\theta)} & \cos (\theta) & =\frac{1}{\sec (\theta)} \\ \cot (\theta) & =\frac{1}{\tan (\theta)} & \tan (\theta) & =\frac{1}{\cot (\theta)} \end{aligned} $ Pythagorean Identities $ \begin{aligned} & \sin ^2(\theta)+\cos ^2(\theta)=1 \\ & \tan ^2(\theta)+1=\sec ^2(\theta) \\ & 1+\cot ^2(\theta)=\csc ^2(\theta) \end{aligned} $ Even/Odd Formulas $ \begin{aligned} \sin (-\theta)&=-\sin (\theta) & \csc (-\theta)&=-\csc (\theta) \\ \cos (-\theta)&=+\cos (\theta) & \sec (-\theta)&=+\sec (\theta) \\ \tan (-\theta)&=-\tan (\theta) & \cot (-\theta)&=-\cot (\theta) \end{aligned} $ Periodic Formulas If $n$ is an integer then, $ \begin{aligned} & \sin (\theta+2 \pi n)=\sin (\theta) \quad \csc (\theta+2 \pi n)=\csc (\theta) \\ & \cos (\theta+2 \pi n)=\cos (\theta) \sec (\theta+2 \pi n)=\sec (\theta) \\ & \tan (\theta+\pi n)=\tan (\theta) \quad \cot (\theta+\pi n)=\cot (\theta) \\ & \end{aligned} $ Degrees to Radians Formulas If $x$ is an angle in degrees and $t$ is an angle in radians then $ \frac{\pi}{180}=\frac{t}{x} \quad \Rightarrow \quad t=\frac{\pi x}{180} \quad \text { and } \quad x=\frac{180 t}{\pi} $ Double Angle Formulas $ \begin{aligned} \sin (2 \theta) & =2 \sin (\theta) \cos (\theta) \\ \cos (2 \theta) & =\cos ^2(\theta)-\sin ^2(\theta) \\ & =2 \cos ^2(\theta)-1 \\ & =1-2 \sin ^2(\theta) \\ \tan (2 \theta) & =\frac{2 \tan (\theta)}{1-\tan ^2(\theta)} \end{aligned} $ Half Angle Formulas $ \begin{aligned} & \sin \left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos (\theta)}{2}} \\ & \cos \left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1+\cos (\theta)}{2}} \\ & \tan \left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos (\theta)}{1+\cos (\theta)}} \end{aligned} $ Half Angle Formulas (alternate form) $ \begin{aligned} & \sin ^2(\theta)=\frac{1}{2}(1-\cos (2 \theta)) \\ & \cos ^2(\theta)=\frac{1}{2}(1+\cos (2 \theta)) \end{aligned} \tan ^2(\theta)=\frac{1-\cos (2 \theta)}{1+\cos (2 \theta)} $ Sum and Difference Formulas %%MAKE DIAGRAM%% $ \begin{aligned} & \sin (\alpha \pm \beta)=\sin (\alpha) \cos (\beta) \pm \cos (\alpha) \sin (\beta) \\ & \cos (\alpha \pm \beta)=\cos (\alpha) \cos (\beta) \mp \sin (\alpha) \sin (\beta) \\ & \tan (\alpha \pm \beta)=\frac{\tan (\alpha) \pm \tan (\beta)}{1 \mp \tan (\alpha) \tan (\beta)} \end{aligned} $ Product to Sum Formulas $ \begin{aligned} & \sin (\alpha) \sin (\beta)=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)] \\ & \cos (\alpha) \cos (\beta)=\frac{1}{2}[\cos (\alpha-\beta)+\cos (\alpha+\beta)] \\ & \sin (\alpha) \cos (\beta)=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)] \\ & \cos (\alpha) \sin (\beta)=\frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)]\end{aligned}$ Sum to Product Formulas $\begin{aligned} \sin (\alpha)+\sin (\beta)&=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right) \\ \sin (\alpha)-\sin (\beta)&=2 \cos \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right) \\ \cos (\alpha)+\cos (\beta)&=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right) \\ \cos (\alpha)-\cos (\beta)&=-2 \sin \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right) \end{aligned} $ Cofunction Formulas $ \begin{aligned} \sin \left(\frac{\pi}{2}-\theta\right)&=\cos (\theta) & \cos \left(\frac{\pi}{2}-\theta\right)&=\sin (\theta) \\ \csc \left(\frac{\pi}{2}-\theta\right)&=\sec (\theta) & \sec \left(\frac{\pi}{2}-\theta\right)&=\csc (\theta) \\ \tan \left(\frac{\pi}{2}-\theta\right)&=\cot (\theta) & \cot \left(\frac{\pi}{2}-\theta\right)&=\tan (\theta) \end{aligned} $ Multiple Angle Identities $\begin{aligned} & \sin (2 x)=2 \cos x \sin x \\ & \sin (3 x)=3 \cos ^2 x \sin x-\sin ^3 x = 3\sin x-4\sin^3 x\\ & \sin (4 x)=4 \cos ^3 x \sin x-4 \cos x \sin ^3 x \\ & \sin (5 x)=5 \cos ^4 x \sin x-10 \cos ^2 x \sin ^3 x+\sin ^5 x \\ & \cos (2 x)=\cos ^2 x-\sin ^2 x =1-2 \sin ^2 x = -1+2 \cos ^2 x\\ & \cos (3 x)=\cos ^3 x-3 \cos x \sin ^2 x = \cos x\left(1-4 \sin ^2 x\right)=-3 \cos x+4 \cos ^3 x \\ & \cos (4 x)=\cos ^4 x-6 \cos ^2 x \sin ^2 x+\sin ^4 x=1-8 \sin ^2 x+8 \sin ^4 x \\ & \cos (5 x)=\cos ^5 x-10 \cos ^3 x \sin ^2 x+5 \cos x \sin ^4 x=\cos x\left(1-12 \sin ^2 x+16 \sin ^4 x\right)= 5 \cos x-20 \cos ^3 x+16 \cos ^5 x \end{aligned}$