Question

if sec o +tan o =p then prove that sin o =p2-1/p2+1​

Answer 1

Answer:

9. Lime, gravel and carbonate are found in red soil.

Answer 2

Answer:

Formula used :–

$\pink{\boxed{ \green{ \rm({a} + b)^{2} = {a}^{2} + {b}^{2} + 2ab}}}$

$\blue{ \boxed{{ \rm \orange{{sec}^{2} \theta - 1 = {tan}^{2} \theta } }}}$

$\red{\boxed{ \purple{\tan^{2} \theta + 1 = { \sec}^{2} \theta}}}$

$\green{\boxed{ \pink{\frac{1}{ \cos \theta} = \sec\theta}}}$

$\orange{\boxed{ \blue{\tan \theta = \frac{ \sin \theta}{ \cos \theta}}}}$

Step-by-step explanation:

$\rm \scriptsize \: if \: sec \theta +tan \theta=p \: then \: prove \: that \: sin \theta = \frac{ {p}^{2} - 1 }{ {p}^{2} + 1}$

Have Given :-

$\rm sec \theta +tan \theta=p$

$\rm \frac{ {p}^{2} - 1 }{ {p}^{2} + 1}$

$\rm = \frac{ {(sec \theta} + tan \theta)^{2} - 1 }{{(sec \theta} + tan \theta)^{2} + 1}$

$\rm = \frac{ {sec}^{2} \theta + {tan}^{2} \theta + 2sec \theta.tan \theta - 1 }{{sec}^{2} \theta + {tan}^{2} \theta + 2sec \theta.tan \theta + 1} \: [ \because \: ({a} + b)^{2} = {a}^{2} + {b}^{2} + 2ab ]$

$\rm = \frac{( {sec}^{2} \theta- 1 )+ {tan}^{2} \theta + 2sec \theta.tan \theta }{{sec}^{2} \theta +( {tan}^{2} \theta+ 1) + 2sec \theta.tan \theta }$

$\scriptsize \rm = \frac{ {tan}^{2} \theta + {tan}^{2} \theta + 2sec \theta.tan \theta }{{sec}^{2} \theta + {sec}^{2} \theta + 2sec \theta.tan \theta } \: [ \because \: {sec}^{2} \theta - 1 = {tan}^{2} \theta \: and \: \tan^{2} \theta + 1 = {sec}^{2} \theta]$

$\rm = \frac{2 {tan}^{2} \theta + 2sec \theta.tan \theta }{2{sec}^{2} \theta + 2sec \theta.tan \theta } \:$

$\rm = \frac{2 {tan}^{2} \theta + 2 \tan \theta.sec \theta }{2{sec}^{2} \theta + 2sec \theta.tan \theta } \:$

$\rm = \frac{2 {tan \theta} \cancel{(\tan \theta + sec \theta )} }{2{sec} \theta \cancel{(sec \theta + tan \theta )} }$

$\rm = \frac{ \cancel2 {tan \theta}}{ \cancel2{sec} \theta }$

$\rm = \frac{ {tan \theta}}{ {sec} \theta }$

$\rm = \frac{ {tan \theta}}{ \frac{1}{ \cos\theta}} \: \: \: [ \because \: \frac{1}{cos \theta} = sec\theta]$

$\rm = \tan \theta \times cos \theta$

$\rm = \frac{sin \theta}{ \cancel{cos \theta}} \times \cancel{cos \theta} \: \: [ \because \: tan \theta = \frac{sin \theta}{ \cos \theta} ]$

$\rm = sin \theta$

Hance Proved