Question

o Prove that ;

1/1+sin2theta + 1/1+cos2theta +1/1+sec2theta +1/1+cosec2theta = 2​

Answer 1

Answer:

Given that:-

$\tt \red{ \frac{1}{1 + { \sin }^{2} θ} + \frac{1}{1 + { \cos}^{2} θ} + \frac{1}{1 + { \sec }^{2} θ} + \frac{1}{1 + \csc^{2} θ} = \green 2 }$

What to do:-

To prove LHS = RHS.

Solution:-

$\tt \red{ \frac{1}{1 + { \sin }^{2} θ} + \frac{1}{1 + { \cos}^{2} θ} + \frac{1}{1 + { \sec }^{2} θ} + \frac{1}{1 + \csc^{2} θ} = \green 2 }$

$⟹ \tt \red{\frac{1}{1 + { \sin}^{2}θ } + \frac{1}{1 + { \cos }^{2} θ} + \frac{1}{1 + \frac{1}{ { \cos}^{2} θ } } + \frac{1}{1 + \frac{1}{ { \sin }^{2} θ} }} = \green2$

$⟹ \tt \red {\frac{1}{1 + { \sin }^{2}θ } + \frac{1}{1 + { \cos }^{2} θ} + \frac{ { \cos }^{2} θ}{1 + \frac{1}{ { \cos }^{2}θ } } + \frac{ { \sin }^{2}θ }{1 + \frac{1}{ { \sin }^{2} θ} } } = \green 2$

$⟹ \tt \red{\frac{1}{1 + { \sin }^{2}θ } + \frac{1}{1 { \cos }^{2} θ} + \frac{1}{ { \cos }^{2}θ + 1 } + \frac{1}{ { \sin }^{2} θ + 1} } = \green2$

$⟹ \tt \red{\frac{1}{1 + { \sin }^{2} θ} + \frac{ { \sin }^{2}θ }{1 + { \sin }^{2}θ } + \frac{1}{1 + { \cos }^{2}θ } + \frac{ { \cos }^{2} θ}{1 + { \cos }^{2} θ} } = \green2$

$⟹ \tt \red {\frac{1 + { \sin }^{2} θ}{1 + { \sin }^{2} θ} + \frac{1 + { \cos }^{2} θ}{1 + { \cos }^{2} θ}} = \green2$

$⟹1 + 1 = \green2$

$⟹ \red2 = \green2$

LHS= RHS

Hence Proved.