Limits using Trig Identities

Refer to Trigonometry Identities to solve these problems

Example (Pythagorean Identity) §

lim $\frac{1-cos\theta }{2si{n}^2\theta }$ = $\frac{1-cos\theta }{2(1-co{s}^2\theta )}$ = $\frac{1-cos\theta }{2(1+cos\theta )(1-cos\theta )}$ = $\frac{1}{2+2cos\theta }$ , $\theta$ $\ne$ 0 $\theta$ -> 0

lim $\frac{1}{2+2cos(0)}$ = $\frac{1}{2+2}$ = $\frac{1}{4}$ $\theta$ -> 0

Example (Double Angle Identity) §

lim $\frac{1+\sqrt{2}sin\theta }{cos2\theta }$ = $\frac{1+\sqrt{2}sin\theta }{(1+\sqrt{2}sin\theta )(1-\sqrt{2}sin\theta )}$ = $\frac{1}{1-\sqrt{2}sin\theta }$ , $\theta$ $\ne$ $\frac{-\pi }{4}$ $\theta$ -> -$\frac{\pi }{4}$

lim $\frac{1}{1-\sqrt{2}sin(\frac{-\pi }{4})}$ = $\frac{1}{1+1}$ = $\frac{1}{2}$ $\theta$ -> $\frac{-\pi }{4}$ **

Worked Examples §