Question

trigonometry prove that

Answer 1

we know,
sec^2@ - tan^2@ = 1 by formula .
we also know by algebric formula ,
a^2 - b^2 = (a - b)( a + b)

use this ,
sec^2@ - tan^2 @ = 1 = (sec @ - tan@)(sec@ + tan@ )

so,
sec@ - tan@ = 1/( sec@ + tan@ )
hence proved

Answer 2

LHS
= 1/secФ - tanФ

We know sec²Ф - tan²Ф = 1

So we have to rationalize the denominator.
= 1(secФ + tanФ) / (secФ - tanФ)(secФ + tanФ)
= secФ + tanФ / sec²Ф - tan²Ф  [ (a + b)(a - b) = a² - b² ]
= secФ + tanФ / 1        (from above mentioned trigonometry identity)
= secФ + tanФ

Hence Proved

Hope This Helps You!