Lim x-4 1-sinxcosxcos2x-sin2x- 1-2sinx

1 sin xcos xcos^2 x sin ^2 x=1 tan xcos 2 x=1 tan x1 tan ^2 x1+tan ^2 x=(1 tan x)(1+tan^2 x)(1 tan x)(1+tan x)=1+tan^2 x1+tan x lim _x→π/41 sin xcos x ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Limits

lim x→π/4 1-s...

Question

$lim x \to π / 4 \frac{1 - \frac{sin x}{cos x}}{{cos}^{2} x - {sin}^{2} x} \times (1 + \sqrt{2} sin x)$

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Solution

$\frac{1 - \frac{sin x}{cos x}}{{cos}^{2} x - {sin}^{2} x} = \frac{1 - tan x}{cos 2 x} = \frac{1 - tan x}{\frac{1 - {tan}^{2} x}{1 + {tan}^{2} x}} = \frac{(1 - tan x) (1 + {tan}^{2} x)}{(1 - tan x) (1 + tan x)} = \frac{1 + {tan}^{2} x}{1 + tan x}$
$lim x \to π / 4 \frac{1 - \frac{sin x}{cos x}}{{cos}^{2} x - {sin}^{2} x} \times (1 + \sqrt{2} sin x) = lim x \to π / 4 \frac{1 + {tan}^{2} x}{1 + tan x} \times (1 + \sqrt{2} sin x) = \frac{1 + 1}{1 + 1} \times (1 + \sqrt{2} \times \frac{1}{\sqrt{2}}) = 2$

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limx→π/41−sinxcosxcos2x−sin2x×(1+√2sinx)

1−sinxcosxcos2x−sin2x=1−tanxcos2x=1−tanx1−tan2x1+tan2x=(1−tanx)(1+tan2x)(1−tanx)(1+tanx)=1+tan2x1+tanxlimx→π/41−sinxcosxcos2x−sin2x×(1+√2sinx)=limx→π/41+tan2x1+tanx×(1+√2sinx)=1+11+1×(1+√2×1√2)=2

$lim x \to π / 4 \frac{1 - \frac{sin x}{cos x}}{{cos}^{2} x - {sin}^{2} x} \times (1 + \sqrt{2} sin x)$