solve the equation tan 3θcotθ+1=0 for 0≤θ≤180 b)show that if cos 2θ is not zero then cos 2θ+sec 2θ=2[((cos^4 θ+sin^4 θ)/(cos^4 θ−sin^4 θ))] c)find the limit of ((tan (θ/3))/(3θ)) as θ→0


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Question Number 50892 by peter frank last updated on 21/Dec/18

$s o l v e t h e e q u a t i o n$ $\tan 3 θ c o t θ + 1 = 0 f o r$ $0 ⩽ θ ⩽ 180$ $b) s h o w t h a t i f \cos 2 θ i s n o t z e r o$ $t h e n$ $\cos 2 θ + \sec 2 θ = 2 [\frac{\cos^{4} θ + \sin^{4} θ}{\cos^{4} θ - \sin^{4} θ}]$ $c) f i n d t h e l i m i t o f$ $\frac{\tan \frac{θ}{3}}{3 θ} a s θ \to 0$
	Answered by kaivan.ahmadi last updated on 21/Dec/18

	$\frac{\cot θ}{\cot 3 θ} + 1 = 0 \Rightarrow \frac{\cot θ}{\cot 3 θ} = - 1 \Rightarrow \cot θ = - \cot 3 θ$ $\Rightarrow \cot θ = \cot (- 3 θ)$ $θ = k π - 3 θ \Rightarrow θ = \frac{k π}{4}$ $θ = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ $2 (\frac{\cos^{4} θ + \sin^{4} θ}{\cos^{4} θ - \sin^{4} θ}) = 2 (\frac{{(\cos^{2} θ + \sin^{2} θ)}^{2} - {2 \sin}^{2} θ \cos^{2} θ}{(\cos^{2} θ - \sin^{2} θ) (\cos^{2} θ + \sin^{2} θ)}) =$ $\frac{2 - \sin^{2} 2 θ}{\cos 2 θ} = \frac{(1 - \sin^{2} 2 θ) + 1}{\cos 2 θ} = \frac{\cos^{2} 2 θ + 1}{\cos 2 θ} = \cos 2 θ + \sec 2 θ$
	Answered by peter frank last updated on 21/Dec/18

	$\frac{\sin 3 θ \cos θ}{\cos 3 θ \sin θ} + 1 = 0$ $\frac{\sin 3 θ \cos θ + \cos 3 θ \sin θ}{\cos 3 θ \sin θ} = 0$ $\frac{\sin 4 θ}{\cos 3 θ \sin θ} = 0$ $θ = 45 a n d 135$
	Answered by peter frank last updated on 21/Dec/18

	$\lim_{x \to 0} \frac{\sin \frac{θ}{3}}{\frac{\cos \frac{θ}{3}}{3 θ} .}$ $\lim_{x \to 0} \frac{\sin \frac{θ}{3}}{\frac{\cos \frac{θ}{3}}{3 θ (\frac{3}{3})} .} q$ $\lim_{x \to 0} \frac{\sin \frac{θ}{3}}{\frac{θ}{3} . 9} . \frac{1}{\cos \frac{θ}{3}} = \lim_{x \to 0} 1 . \frac{1}{9} = \frac{1}{9}$ $\frac{1}{9}$
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