Question

if cos a= 5/13 find sin^2a+1/sec^2a

Answer 1

Solution:-

Given

$\rm \to \cos a = \dfrac{5}{13}$

To find

$\rm \to \: \sin ^{2} a + \dfrac{1}{ \sec ^{2} a}$

$\rm \to \: \sin^{2} a + \dfrac{1}{ \dfrac{1}{ \cos ^{2}a } }$

$\rm \to \: \sin^{2} a + \cos ^{2} a$

Now Take

$\rm \to \cos a = \dfrac{5}{13} = \dfrac{b}{h}$

Using Pythagoras theorem

$\rm \to \: b = 5,h = 13 \: and \: p = x$

$\rm \to \: {h}^{2} = {p}^{2} + {b}^{2}$

$\rm \to {13}^{2} = {x}^{2} + {5}^{2}$

$\rm \to \: 169 - 25 = {x}^{2}$

$\rm \to \: x = \sqrt{144} = 12$

$\rm \to \: b = 5,h = 13 \: and \: p = 12$

Put the value on equation

$\rm \to \: \sin^{2} a + \cos ^{2} a$

$\rm \to \: \bigg( \dfrac{12}{13} \bigg)^{2} + \bigg( \dfrac{5}{13} \bigg)^{2}$

$\rm \to \: \dfrac{144}{169} + \dfrac{25}{169}$

$\rm \to \: \dfrac{169}{169} = 1$

Answer is 1