Question

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Question for you :-

★ PROVE :
(secø - cosø)(cotø+tanø) = tanø secø

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Answer 1

To Prove :-

$\sf (sec \theta - cos \theta )(cot \theta + tan \theta ) = tan \theta + sec \theta$

Solution :-

$\sf L.H.S = (sec \theta - cos \theta )( cot \theta + tan \theta )$

$\bullet \bf \ sec \theta = \dfrac{1}{cos \theta } \\\\\bullet \bf \ cot \theta = \dfrac{cos \theta }{sin \theta } \\\\\bullet \bf \ tan \theta = \dfrac{sin \theta }{cos \theta }$

        $= \sf \left( \dfrac{1}{cos \theta } - cos \theta \right) \left( \dfrac{cos \theta }{sin \theta } +\dfrac{sin \theta }{cos \theta } \right) \\\\= \left( \dfrac{1 - cos^2 \theta }{cos\theta } \right) \left( \dfrac{cos^2 \theta + sin^2 \theta }{sin \theta cos \theta } \right)$

$\bullet \bf \ cos^2 \theta + sin^2 \theta = 1$

$\bullet \bf \ 1- cos^2 \theta = sin^2 \theta$

          $= \sf \dfrac{sin^2 \theta } {cos \theta } \times \dfrac{1}{sin \theta cos \theta }\\\\= \dfrac{sin \theta }{cos^2 \theta } \\\\= \dfrac{sin \theta }{cos \theta } \times \dfrac{1}{cos \theta }\\\\= tan \theta sec \theta \\\\= R.H.S$

Hence proved.

Answer 2

Given :

• (secø - cosø)(cotø+tanø) = tanø secø

To Prove :

• (secø - cosø)(cotø+tanø) = tanø secø

Solution :

$\\ \sf \: : \implies \: \: \: \: \: \: \: \: { \large{( \sec(a) - \cos(a) )( \cot(a) + \tan(a) = \tan(a) \times \sec(a) ) }} \\ \\ \\ \sf \: : \implies \: \: \: \: \: \: \: \: { \large{( \frac{1}{ \cos(a) } - \cos(a) )( \frac{ \cos(a) }{ \sin(a) } + \frac{ \sin(a) }{ \cos(a) } )}} \\ \\ \\ \sf \: : \implies \: \: \: \: \: \: \: \:{ \large{ \frac{1 - { \cos {}^{2} (a) }^{} }{ \cos(a) } \times \frac{ \cos {}^{2} (a) + \sin {}^{2} (a) }{ \cos(a) \times \sin(a) } }} \\ \\ \\ \sf \: : \implies \: \: \: \: \: \: \: \:{ \large{ \frac{ \sin {}^{2} (a) }{ \cos(a) } \times \frac{1}{ \cos(a) \sin(a) } }} \\ \\ \\ \\ \sf \: : \implies \: \: \: \: \: \: \: \:{ \large{ \frac{ \sin(a) }{ \cos(a) } \times \frac{1}{ \cos(a) } }} \\ \\ \\ \sf \: : \implies \: \: \: \: \: \: \: \:{ \large{ \tan(a) \times \sec(a) }}$

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