Question

if sec x + tan x =p find cosec x

Answer 1

sec x + tan x = p
Let's multiply and divide by sec x - tan x in LHS

We get,
sec^2 x - tan^2 x / sec x - tan x = p
Therefore,
1 / sec x - tan x = p
Since,
sec^2 x - tan^2 x = 1 from trigonometry

Now,
sec x + tan x = p
and sec x - tan x = 1/p

Adding both
2sec x = p^2 + 1/p
sec x = p^2 + 1 / 2p
cos x = 2p/p^2 + 1

Now,
sin x = √ 1- cos ^2 x = √ 1-4p^2 / p^4 + 2p^2 + 1
Simplifying,

sin x = √ (p^2 - 1)^2 / (p^2 + 1)^2
Therefore,
sin x = p^2 - 1/p^2 + 1.

Answer 2

Answer:

secA+tanA=p ----------------------------(1)

∵, sec²A-tan²A=1

or, (secA+tanA)(secA-tanA)=1

or, secA-tanA=1/p -----------------------(2)

Subtracting (2) from (1) we get,

2tanA=p-1/p

or, tanA=(p²-1)/2p

∴, cotA=2p/(p²-1)

Now, cosec²A-cot²A=1

or, cosec²A=1+cot²A

or, cosec²A=1+{2p/(p²-1)}²

or, cosec²A=1+4p²/(p²-1)²

or, cosec²A=(p⁴-2p²+1+4p²)/(p²-1)²

or, cosec²A=(p⁴+2p²+1)/(p²-1)²

or, cosec²A=(p²+1)²/(p²-1)²

or, cosecA=(p²+1)/(p²-1)

I THINK IT MAY BE BIT DIFFICULT TO UNDERSTAND...