Question

If :
sec A + tan A = x
Show that :
sin A = (x^2 - 1) / (x^2 + 1)

Answer 1

1 is the answer because sec and tan are recipoxals of each other

Answer 2

Hope u like my process
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We know,

sec²A - tan² A = 1

secA + tanA = x ____(1)

So,

sec²A - tan²A = 1

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

or, (secA - tanA)* x = 1

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

Adding eq (1) and (2) we get,
__________________
=> secA + tanA = x
+ secA - tanA = 1/x
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=> 2 secA = (x +1/x) = (x² +1)/x
___________________
Subtracting eq (1) and (2) we get,
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_secA + tanA = x
- (secA - tanA) = - 1/x
_________________
=> 2tanA = x - 1/x = (x² - 1)/x
————————————
Now,

$\sin(a) = \frac{ \tan( a) }{ \sec(a) } = \frac{2 \tan(a) }{2 \sec(a) } \\ = \frac{ \frac{ { x}^{2} - 1}{x} }{ \frac{ {x}^{2} + 1 }{x} } = \frac{ {x}^{2} - 1}{ {x}^{2} + 1}$



Hence..... proved.

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