Question

If tan x=t,then tan 2x + sec 2x is equal to
a)1+t/1-t
b)1-t/1+t
c)2t/1-t
d)2t/1+t​

Answer 1

$Given \: tan x = t \: ---(1)$

$tan 2x + sec 2x \\= \frac{sin 2x}{ Cos 2x } + \frac{1}{ Cos 2x } \\= \frac{sin 2x + 1 }{ cos 2x } \\= \frac{ cos^{2} x + sin^{2} x + 2sin x cos x }{ cos^{2} x - sin^{2} x }$

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/* We know that */

1. Cos² x + Sin² x = 1

2. Sin 2x = 2sinx cosx

3. cos 2x = cos²x - sin²x

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$= \frac{ ( Cos x + sin x )^{2} } { (cosx + sin x )( Cos x - sin x ) } \\= \frac{ cos x + sin x }{ cos x - sin x } \\= \frac{ \frac{cos x}{cos x} + \frac{sin x }{cos x } }{ \frac{cos x}{cos x} - \frac{sin x }{cos x } }\\= \frac{ 1 + tan x }{ 1 - tan x} \\\green {= \frac{ 1 + t }{ 1 - t } }$

Therefore.,

$Option \: \pink{ ( a ) } \: is \: correct.$

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