Question

Prove cot a- tan a= 2cos^2a-1/sina cosa

Answer 1

$\huge\sf\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf\orange{To \ prove:}$

$\sf{cot \ a \ - \ tan \ a=\frac{2cos^{2}a-1}{sin \ a \ cos \ a}}$

$\sf\green{\underline{\underline{Proof:}}}$

$\sf{L.H.S.=cot \ a \ - \ tan \ a}$

$\sf{(cot=\frac{cos}{sin})}$

$\sf{(tan=\frac{sin}{cos})}$

$\sf{Trigonometric \ ratio}$

$\sf{=\frac{cos \ a}{sin \ a}-\frac{sin \ a}{cos \ a}}$

$\sf{=\frac{cos^{2}a-sin^{2}a}{sin \ a \ cos \ a}}$

$\sf{sin^{2}=1-cos^{2}}$

$\sf{Trigonometric \ identity}$

$\sf{=\frac{cos^{2}a-(1-cos^{2}a)}{sin \ a \ cos \ a}}$

$\sf{=\frac{cos^{2}a-1+cos^{2}a}{sin \ a \ cos \ a}}$

$\sf{=\frac{2cos^{2}a-1}{sin \ a \ cos \ a}}$

$\sf{=R.H.S.}$

$\sf{Hence, \ proved}$

$\sf\purple{cot \ a \ - \ tan \ a=\frac{2cos^{2}a-1}{sin \ a \ cos \ a}}$

Answer 2

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

• ✦ we need to prove that cot a - tan a = 2cos² a - 1/sin a. cos a

$\huge{\underline{\bf{\red{Solution:-}}}}$

LHS :-

$\longmapsto \rm\:\:cot \: a \: - tan \: a \: \\ \\ \rm\:\:we \: know \: that \:{ \blue{ cot \theta = \frac{cos \theta}{sin \theta} }}\: \\ \\ \longmapsto \rm\:\: \frac{cos \: a}{sin \: a} - tan \: a \\ \\ \rm\:\: we \: know \: that \: { \blue{tan \theta = \frac{sin \theta}{cos \theta} }} \\ \\\longmapsto \rm\:\: \frac{cos \: a}{sin \: a} - \frac{sin \:a }{cos \ \: a} \\ \\ \longmapsto \rm\:\: \frac{ {cos}^{2} a - {sin}^{2}a }{sin \: a.cos \: a} \\ \\ \rm{ \blue{sin {}^{2} \theta = 1 - cos {}^{2} \theta }} \:\: \\ \\\longmapsto \rm\:\: \frac{ {cos}^{2} a - (1 - cos {}^{2} a)}{sin \: a.cos \: a} \\ \\ \longmapsto \rm\:\: \frac{ {cos}^{2} a - 1 + {cos}^{2} a}{sin \: a .cos \: a} \\ \\\longmapsto \bf\:\: \frac{2 {cos}^{2} a - 1}{sin \: a.cos \: a}$

LHS = RHS

Hence proved.

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