Question

cot theta +tan theta =co sec theta sec theta​

Answer 1

Identies used:-

• Sin²Ø+Cos²Ø = 1

• 1/CosØ = SecØ

• 1/SinØ = CosecØ

Solution:-

$\implies\cot( \theta) + \tan( \theta) \\ \\\implies \frac{ \cos( \theta) }{ \sin( \theta) } + \frac{ \sin( \theta) }{ \cos( \theta) } \\ \\ \implies\frac{\cos ^{2} ( \theta) + \sin ^{2} ( \theta) }{ \sin( \theta) \cos( \theta) } \\ \\ \implies\frac{1}{ \sin( \theta) \cos( \theta) } \\ \\ \implies\cos( \theta) \sec( \theta)$

Answer 2

Given : $\star \bf \cot \theta + \tan \theta = \cosec \theta \sec \theta$

Exigency To Prove : $\longmapsto \sf \cot \theta + \tan \theta = \cosec \theta \sec \theta$

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$\qquad \star\:\: \bf \cot \theta + \tan \theta = \cosec \theta - \sec \theta$

Here ,

• $\star \bf {L.H.S} = \sf \cot \theta + \tan \theta$
• $\star \bf {R.H.S} = \sf \cosec \theta \sec \theta$

⠀⠀⠀⠀⠀⠀$\underline {\frak{\star\:Now \: By \: Solving \: the \: L.H.S \: \::}}$

• $\star \bf {L.H.S} = \sf \cot \theta + \tan \theta$

$:\implies\cot \theta + \tan \theta \\ \\ \sf{As,\;We\:know\:that\::}\\\\ \star \tan \theta = \dfrac{\sin \theta }{\cos \theta }\\\\ \bf And :\\\\\star \cot \theta = \dfrac{\cos \theta }{\sin \theta }\\\\ :\implies \dfrac{ \cos \theta }{ \sin \theta } + \frac{ \sin \theta }{ \cos \theta } \\ \\ \implies\frac{\cos ^{2} \theta + \sin ^{2} \theta }{ \sin \theta \cos \theta }\\\\ \sf{As,\;We\:know\:that\::}\\\\ \star sin^2\theta + \cos^2 \theta = 1 \\ \\ :\implies\frac{1}{ \sin \theta \cos \theta } \\ \\:\implies \dfrac{1}{\sin\theta} \dfrac{1}{\cos \theta } \\\\ :\implies\bf L.H.S = \cosec \theta \sec \theta$

Therefore,

• $\star \bf {L.H.S} = \sf \cosec \theta \sec \theta$
• $\star \bf {R.H.S} = \sf \cosec \theta \sec \theta$

As , We can see that ,

• L.H.S = R.H.S

⠀⠀⠀⠀⠀$\therefore {\underline {\bf{ Hence, \:Verified \:}}}$

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$\large {\boxed{\sf{\mid{\overline {\underline {\star More\:To\:know\::}}}\mid}}}$

$\boxed{\begin{array}{cc} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{array}}$

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