Question

If ; sec x + tan x = p obtain the values of sec x, tan x and sin x in terms of p.

Answer 1

Given : secx + tanx = p ................................... (i)
Now, we know that : sec²x - tan²x = 1
or, (secx + tanx)(secx - tanx) = 1

Putting (i) in the equation, we get :-
p(secx - tanx) = 1
or, secx - tanx = 1/p .........................................(ii)

Now adding (i) and (ii), we get :-
secx + tanx + secx - tanx = 1/p + p
or, 2secx = (1+p²)/p
or, secx = (1+p²)/2p   [Ans 1]

On subtracting (ii) from (i), we get :-
(secx + tanx) - (secx - tanx) = p - 1/p
or, 2tanx = (p² - 1)/p
or, tanx = (p² - 1)/2p    [Ans 2]

We know that sinx = sinx/cosx x cosx
or, sinx = tanx x 1/secx
or, sinx = tanx/secx
or, sinx = [(p² - 1)/2p]/[(p² + 1)/2p] ..(In the next step both the 2p get cancelled)
or, sinx = (p² - 1)/(p² + 1)    [Ans 3]

Hope that helps !!

Answer 2

Given : secx + tanx = p ................................... (i)

Now, we know that : sec²x - tan²x = 1

or, (secx + tanx)(secx - tanx) = 1

Putting (i) in the equation, we get :-

p(secx - tanx) = 1

or, secx - tanx = 1/p .........................................(ii)

Now adding (i) and (ii), we get :-

secx + tanx + secx - tanx = 1/p + p

or, 2secx = (1+p²)/p

or, secx = (1+p²)/2p [Ans 1]

On subtracting (ii) from (i), we get :-

(secx + tanx) - (secx - tanx) = p - 1/p

or, 2tanx = (p² - 1)/p

or, tanx = (p² - 1)/2p [Ans 2]

We know that sinx = sinx/cosx x cosx

or, sinx = tanx x 1/secx

or, sinx = tanx/secx

or, sinx = [(p² - 1)/2p]/[(p² + 1)/2p] ..(In the next step both the 2p get cancelled)

or, sinx = (p² - 1)/(p² + 1) [Ans 3]