Question

prove that
sin theta ( 1+tan theta )+cos theta ( 1+cot theta) is equal to sec theta + cosec theta

Answer 1

plz find the the attachment

Answer 2

Answer:

To Prove: $sin\,\theta(1+tan\,\theta)+cos\,\theta(1+cot\,\theta)=sec\,\theta+cosec\,\theta$

Consider,

$LHS=sin\,\theta(1+tan\,\theta)+cos\,\theta(1+cot\,\theta)$

$=sin\,\theta(1+\frac{sin\,\theta}{cos\,\theta})+cos\,\theta(1+\frac{cos\,\theta}{sin\,\theta})$

$=sin\,\theta(\frac{cos\,\theta+sin\,\theta}{cos\,\theta})+cos\,\theta(\frac{sin\,\theta+cos\,\theta}{sin\,\theta})$

$=(cos\,\theta+sin\,\theta)(\frac{sin\,\theta}{cos\,\theta})+\frac{cos\,\theta}{sin\,\theta})$

$=(cos\,\theta+sin\,\theta)(\frac{sin^2\,\theta+cos^2\,\theta}{cos\,\theta\:sin\,\theta})$

$=(cos\,\theta+sin\,\theta)(\frac{1}{cos\,\theta\:sin\,\theta})$

$=\frac{cos\,\theta+sin\,\theta}{cos\,\theta\:sin\,\theta}$

$=\frac{cos\,\theta}{cos\,\theta\:sin\,\theta}+\frac{sin\,\theta}{cos\,\theta\:sin\,\theta}$

$=\frac{1}{sin\,\theta}+\frac{1}{cos\,\theta}$

$=cosec\,\theta+sec\,\theta$

= RHS

Hence Proved.