1. Problem: Write each expression as a trigonometric function of a single angle. (i) sin 37° cos 22° + cos 37° sin 22° - Use the sine addition formula: $$\sin(A+B) = \sin A \cos B + \cos A \sin B$$ - Here, $$A=37^\circ$$ and $$B=22^\circ$$ - So, $$\sin 37^\circ \cos 22^\circ + \cos 37^\circ \sin 22^\circ = \sin(37^\circ + 22^\circ) = \sin 59^\circ$$ (ii) cos 83° cos 53° + sin 83° sin 53° - Use the cosine addition formula: $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$ - Here, $$A=83^\circ$$ and $$B=53^\circ$$ - So, $$\cos 83^\circ \cos 53^\circ + \sin 83^\circ \sin 53^\circ = \cos(83^\circ - 53^\circ) = \cos 30^\circ$$ (iii) cos 19° cos 5° – sin 19° sin 5° - Use the cosine addition formula: $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$ - Here, $$A=19^\circ$$ and $$B=5^\circ$$ - So, $$\cos 19^\circ \cos 5^\circ - \sin 19^\circ \sin 5^\circ = \cos(19^\circ + 5^\circ) = \cos 24^\circ$$ (iv) sin 40° cos 15° – cos 40° sin 15° - Use the sine subtraction formula: $$\sin(A-B) = \sin A \cos B - \cos A \sin B$$ - Here, $$A=40^\circ$$ and $$B=15^\circ$$ - So, $$\sin 40^\circ \cos 15^\circ - \cos 40^\circ \sin 15^\circ = \sin(40^\circ - 15^\circ) = \sin 25^\circ$$ (v) $$\frac{\tan 20^\circ + \tan 32^\circ}{1 - \tan 20^\circ \tan 32^\circ}$$ - Use the tangent addition formula: $$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$ - Here, $$A=20^\circ$$ and $$B=32^\circ$$ - So, expression equals $$\tan(20^\circ + 32^\circ) = \tan 52^\circ$$ (vi) $$\frac{\tan 35^\circ - \tan 12^\circ}{1 + \tan 35^\circ \tan 12^\circ}$$ - Use the tangent subtraction formula: $$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$ - Here, $$A=35^\circ$$ and $$B=12^\circ$$ - So, expression equals $$\tan(35^\circ - 12^\circ) = \tan 23^\circ$$ Final answers for 1: (i) $$\sin 59^\circ$$ (ii) $$\cos 30^\circ$$ (iii) $$\cos 24^\circ$$ (iv) $$\sin 25^\circ$$ (v) $$\tan 52^\circ$$ (vi) $$\tan 23^\circ$$