$\text{(a)}\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$

$\text{(b)}\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$

$\text{(c)}{\sin }^{2}A-{\sin }^{2}B={\cos }^{2}B-{\cos }^{2}A=\sin (A+B)\cdot \sin (A-B)$

$\text{(d)}{\cos }^{2}A-{\sin }^{2}B={\cos }^{2}B-{\sin }^{2}A=\cos (A+B)\cdot \cos (A-B)$

$\text{(e)}\tan (A\pm B)=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$

$\text{(f)}\cot (A\pm B)=\frac{\cot A\cot B\mp 1}{\cot B\pm \cot A}$

$\text{(g)}\sin (A+B+C)=\sin A\cos B\cos C+\sin B\cos A\cos C+\sin C\cos A\cos B-\sin A\sin B\sin C$

$\text{(h)}\cos (A+B+C)=\cos A\cos B\cos C-\cos A\sin B\sin C-\sin A\cos B\sin C-\sin A\sin B\cos C$

$\text{(i)}\tan (A+B+C)=\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A}$

(i) $\sin (A+B)+\sin (A-B)=2\sin A\cos B$

    (a) $\sin C+\sin D=2\sin \frac{C+D}{2}\cos \frac{C-D}{2}$

(ii) $\sin (A+B)-\sin (A-B)=2\cos A\sin B$

     (b) $\sin C-\sin D=2\cos \frac{C+D}{2}\sin \frac{C-D}{2}$

(iii) $\cos (A+B)+\cos (A-B)=2\cos A\cos B$

      (c) $\cos C+\cos D=2\cos \frac{C+D}{2}\cos \frac{C-D}{2}$

(iv) $\cos (A-B)-\cos (A+B)=2\sin A\sin B$

(a) $\sin 2A=2\sin A\cos A$

      Note : $\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$ etc.

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

      Note : $2{\cos }^2\frac{\theta }{2}=1+\cos \theta ,2{\sin }^2\frac{\theta }{2}=1-\cos \theta$.

(c) $\tan 2\mathrm{A}=\frac{2\tan \mathrm{A}}{1-{\tan }^2\mathrm{A}}$

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

(d) $\sin 2A=\frac{2\tan A}{1+{\tan }^{2}A},$

      $\cos 2A=\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$

(e) $\sin 3A=3\sin A-4{\sin }^{3}A$

(f) $\cos 3A=4{\cos }^{3}A-3\cos A$

(g) $\tan 3A=\frac{3\tan A-{\tan }^{3}A}{1-3{\tan }^{2}A}$

1. $\sin \theta \cdot \mathrm{cosec}\theta =1$ or, $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$

2. $\cos \theta \cdot \sec \theta =1$ or, $\sec \theta =\frac{1}{\cos \theta }$

3. $\tan \theta \cdot \cot \theta =1$ or, $\cot \theta =\frac{1}{\tan \theta }$

4. $\tan \theta =\frac{\sin \theta }{\cos \theta }$

5. $\cot \theta =\frac{\cos \theta }{\sin \theta }$

6. ${\sin }^2\theta$ implies $(\sin \theta {)}^2;$ similarly, ${\tan }^3\theta$ means $(\tan \theta {)}^3$ etc.

7. ${\sin }^2\theta +{\cos }^2\theta =1;{\cos }^2\theta =1-{\sin }^2\theta ;{\sin }^2\theta =1-{\cos }^2\theta$

8. ${\sec }^2\theta =1+{\tan }^2\theta$ or, ${\sec }^2\theta -{\tan }^2\theta =1$

9. ${\mathrm{cosec}}^2\theta =1+{\cot }^2\theta$ or, ${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$

10. The trigonometric ratios (or functions) of a positive acute angle $\theta$ are always non-negative and

(i) $\sin \theta$ and $\cos \theta$ can never be greater than 1 i.e, $0⩽\sin \theta ⩽1,0⩽\cos \theta ⩽1$

(ii) $\sec \theta$ and $\mathrm{cosec}\theta \mathrm{can}$ never be less than 1 and

(iii) $\tan \theta$ and $\cot \theta$ can have any value.

11. (i) $\sin 30^{\circ }=\cos 60^{\circ }=\frac{1}{2}$

(ii) $\sin 45^{\circ }=\cos 45^{\circ }=\frac{1}{\sqrt{2}}$

(iii) $\sin 60^{\circ }=\cos 30^{\circ }=\frac{\sqrt{3}}{2}$

(iv) $\sin 90^{\circ }=\cos 0^{\circ }=1$

(v) $\sin 0^{\circ }=\cos 90^{\circ }=0$

(vi) $\tan 0^{\circ }=\cot 90^{\circ }=0$

(vii) $\tan 30^{\circ }=\cot 60^{\circ }=\frac{1}{\sqrt{3}}$

(viii) $\tan 45^{\circ }=\cot 45^{\circ }=1$

(ix) $\tan 60^{\circ }=\cot 30^{\circ }=\sqrt{3}$

(x) $\tan 90^{\circ }=\cot 0^{\circ }=$ undefined