Question

guys please help me in solving this question sin theta[1+tan theta] + cos theta [1+ cot theta]=sec theta + cosec theta​

Answer 1

TO PROVE :-

$\\ \sf \: sin\theta(1 + tan\theta) + cos\theta(1 + cot\theta) = sec\theta.cosec \theta$

SOLUTION :-

$\\ \sf \: L.H.S = sin\theta(1 + tan\theta) + cos\theta(1 + cot\theta) \\ \\ \\ \bigstar\boxed{ \bf \: tan\theta = \frac{sin\theta}{cos\theta} } \: \: \: \: \: \: \: \: \: \: \: \: \: \bigstar\boxed{ \bf \:cot\theta = \dfrac{cos\theta}{sin\theta} }$

$\implies \sf \: sin\theta \left( 1 + \dfrac{sin\theta}{cos\theta} \right) + cos\theta \left(1 + \dfrac{cos\theta}{sin\theta} \right)$

$\\ \implies \sf \: sin\theta\left( \dfrac{cos\theta + sin\theta}{cos\theta} \right) + cos\theta\left( \dfrac{sin\theta + cos\theta}{sin\theta} \right)$

$\\ \implies \sf \: \dfrac{sin\theta(cos\theta + sin\theta)}{cos\theta} + \dfrac{cos\theta(sin\theta + cos\theta)}{sin\theta}$

$\\ \implies \sf \: \dfrac{ {sin}^{2}\theta (cos\theta + sin\theta) + {cos}^{2}\theta(sin\theta + cos\theta) }{sin\theta.cos\theta} \\ \\ \\ \bigstar\boxed{\bf \: {sin}^{2}\theta + {cos}^{2}\theta = 1 } \\ \\ \\ \implies\sf \: \dfrac{1}{sin\theta.cos\theta}$

$\\ \\ \bigstar\boxed{ \bf \: \dfrac{1}{sin\theta} = cosec\theta } \: \: \: \: \: \: \: \: \: \: \: \bigstar\boxed{ \bf \: \dfrac{1}{cos \theta} = sec\theta }$

$\\ \implies \sf \: cosec\theta.sec\theta = R.H.S \: \: \: \: \: \: \: (proved)$