section a - simplify using trig identities\na1. simplify: \\( \\frac { 1 - \\cos x } { \\sin x } \\).\na2…

Explanation:

A1.

Step1: Multiply numerator and denominator by (1 + \cos x)

$$\frac{1-\cos x}{\sin x}\times\frac{1 + \cos x}{1+\cos x}=\frac{1-\cos^{2}x}{\sin x(1 + \cos x)}$$

Step2: Use the Pythagorean identity (\sin^{2}x+\cos^{2}x = 1) (i.e., (1-\cos^{2}x=\sin^{2}x))

$$\frac{\sin^{2}x}{\sin x(1 + \cos x)}=\frac{\sin x}{1+\cos x}$$

Step3: Use the double - angle formula (\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}) and (1+\cos x = 2\cos^{2}\frac{x}{2})

$$\frac{2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^{2}\frac{x}{2}}=\tan\frac{x}{2}$$

A2.

Step1: Recall the reciprocal identities (\sec x=\frac{1}{\cos x}) and (\csc x=\frac{1}{\sin x})

$$\sin x\sec x+\cos x\csc x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}$$

Step2: Find a common denominator ((\sin x\cos x))

$$\frac{\sin^{2}x+\cos^{2}x}{\sin x\cos x}$$

Step3: Use the Pythagorean identity (\sin^{2}x+\cos^{2}x = 1)

$$\frac{1}{\sin x\cos x}$$

Step4: Use the double - angle formula (\sin2x = 2\sin x\cos x) (i.e., (\sin x\cos x=\frac{1}{2}\sin2x))

$$\frac{2}{\sin2x}=2\csc2x$$

Answer:

A1. (\tan\frac{x}{2}) A2. (2\csc2x)