Question

Find the general solution 2(sec^2 X +sin^2 X ) =5​

Answer 1

Step-by-step explanation:

We have,

$\tt{2(sec^{2}(x) + sin ^{2} (x)) = 5 }$

$\tt{ \implies2 \bigg( \dfrac{1}{cos^{2}(x)} + sin ^{2} (x) \bigg) = 5 }$

$\tt{ \implies2 \bigg( \dfrac{1 + sin ^{2} (x) \: cos^{2} (x) }{cos^{2}(x)} \bigg) = 5 }$

$\tt{ \implies2 \bigg( \dfrac{1 + \big( 1 - cos ^{2} (x) \big) \: cos^{2} (x) }{cos^{2}(x)} \bigg) = 5 }$

$\tt{ \implies2 \big( 1 + cos^{2} (x) - cos ^{4} (x) \big) = 5 cos^{2}(x)}$

$\tt{ \implies2 + 2cos^{2} (x) -2 cos ^{4} (x) = 5 cos^{2}(x)}$

$\tt{ \implies2 -2 cos ^{4} (x) = 3cos^{2}(x)}$

$\tt{ \implies2 cos ^{4} (x) + 3cos^{2}(x) - 2 = 0}$

$\tt{ \implies2 cos ^{4} (x) + 4cos^{2}(x) - cos^{2}(x) - 2 = 0}$

$\tt{ \implies2 cos ^{2} (x) \big(cos^{2}(x)+ 2 \big) - 1 \big(cos^{2}(x) + 2 \big)= 0}$

$\tt{ \implies \big(2 cos ^{2} (x) - 1 \big)\big(cos^{2}(x)+ 2 \big) = 0}$

$\tt{ \implies \big(2 cos ^{2} (x) - 1 \big) = 0}$

$\tt{ \implies cos ^{2} (x) = \dfrac{1}{2}}$

$\tt{ \implies cos ^{2} (x) = cos^{2} \bigg( \dfrac{ \pi}{4} \bigg)}$

$\sf{ \implies x = n \pi \pm \dfrac{ \pi}{4} }$