Question

If secx + tanx = p and secx - tanx= q then what is value of p square - q square I

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 !!

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Answer 2

Sec x +tan x=p

Sec x - tan x=q

P^2-q^2

=(sec x +tan x) (sec x - tanx)

=sec^2 x-tan ^2 x

=1

Hope it helps☺