Question

simplify 1 + tan square theta into 1 minus sin theta into 1 + sin theta​

Answer 1

Your answer is in the picture.

Answer 2

Given,

$(1 + {tan}^{2} \theta)(1 - \sin \theta)(1 + sin \: \theta)$

Step1:

Identity:

$\huge 1+ {tan}^{2} \theta = { \sec}^{2} \theta$

$So, \: We \: have \: to \: substitute \: {sec}^{2} \theta \: in \: the \: place \: of \: 1 + {tan}^{2} \theta$

Step 2 :

Identity :

$\huge(a + b)(a - b) = {a}^{2} - {b}^{2}$

$(1 - \sin \theta)(1 + sin \: \theta)$

${1}^{2} - { \sin}^{2} \theta$

$1 - { \sin }^{2} \theta$

Identity:

$\huge1 - { \sin }^{2} \theta = { \ \cos }^{2} \theta$

$So \: we \: have \: to \: substitute \: { \cos }^{2} \theta \: in \: the \: place \: of \: (1 - \sin \theta)(1 + \sin \theta)$

Step 3:

$(1 + {tan}^{2} \theta)(1 - \sin \theta)(1 + sin \: \theta)$

${ \sec }^{2} \theta \times { \cos }^{2} \theta$

$1$

$\huge \: Therefore \: (1 + {tan}^{2} \theta)(1 - \sin \theta)(1 + sin \: \theta) = 1$