Question

if sec theta - tan theta = 5 and cosec theta -cot theta =3 then (sec theta+ tan theta)(cosec t
heta +cot theta)=
A)1/5
B)1/3
C)1/15
D)5/3​

Answer 1

Answer:-

Given:-

sec θ - tan θ = 5 -- equation (1)

cosec θ - cot θ = 3 -- equation (2)

We know that,

• sec² θ - tan² θ = 1

using a² - b² = (a + b)(a - b) we get,

⟹ (sec θ + tan θ)(sec θ - tan θ) = 1

⟹ sec θ + tan θ = 1/ sec θ - tan θ

⟹ sec θ + tan θ = 1/5

[ From equation (1) ]

Also,

• cosec² θ - cot² θ = 1

⟹ (cosec θ + cot θ)(cosec θ - cot θ) = 1

⟹ cosec θ + cot θ = 1/ cosec θ - cot θ

⟹ cosec θ + cot θ = 1/3

Now,

We have to find the value of :

⟹ (sec θ + tan θ)(cosec θ + cot θ)

Putting the respective values we get,

⟹ (1/5)(1/3)

⟹ 1/15

∴ The required answer is 1/15 (Option - C).

Answer 2

Answer :-

• $\sf{(sec \theta+ tan \theta)(cosec \theta +cot \theta)=}$$\sf{\red{\dfrac{1}{15}}}$

Given :-

• $\sf{sec\theta -tan\theta = 5}$

• $\sf{cosec\theta -cot\theta = 3}$

To find :-

• $\sf{(sec\theta +tan\theta )(cosec\theta +cot\theta )}$

Formula used :-

• $\boxed{\sf{a²-b²=(a+b)(a-b)}}$

Identity used :-

• $\boxed{\sf{sec²\theta -tan²\theta = 1}}$

• $\boxed{\sf{cosec²\theta -cot²\theta = 1}}$

Solution :-

$\sf{➪\:sec²\theta -tan²\theta = 1}$

$\sf{➪\:(sec\theta +tan\theta)(sec\theta -tan\theta) = 1}$

$\sf{➪\:(sec\theta +tan\theta)(5) = 1}$

$\sf{➪\:sec\theta +tan\theta = \dfrac{1}{5}\:-(i)}$

$\:$

$\sf{➪\:cosec²\theta -cot²\theta = 1}$

$\sf{➪\:(cosec\theta +cot\theta)(cosec\theta -cot\theta) = 1}$

$\sf{➪\:(cosec\theta +cot\theta)(3) = 1}$

$\sf{➪\:cosec\theta +cot\theta = \dfrac{1}{3}\:-(ii)}$

$\:$

By multiplying (i) and (ii),

$\sf{➪\:(sec \theta+ tan \theta)(cosec \theta +cot \theta)}$

$\sf{=(\dfrac{1}{5})(\dfrac{1}{3})}$$=$ $\sf{\red{\dfrac{1}{15}}}$