MathematicsIf tan x =1213, then evaluate the value of 2sin⁡xcos⁡xcos2⁡x-sin2⁡x.

If $\tan x = \frac{12}{13}$ , then evaluate the value of $\frac{2 \sin x \cos x}{\cos^{2} x - \sin^{2} x}$ .

A
312/25
B
350/40
C
150/42
D
450/41

Solution:

Given,

\frac{2 \sin x \cos x}{\cos^{2} x - \sin^{2} x}

…(1)
We know that,

\sin 2 a = 2 \sin a \cos a

and

\cos 2 a = \cos^{2} a - \sin^{2} a

Apply these identities in equation 1,

\frac{2 \sin x \cos x}{\cos^{2} x - \sin^{2} x}

= \frac{\sin 2 x}{\cos 2 x}

= \tan 2 x

= \frac{2 \tan x}{1 - \tan^{2} x}

(Since,

\tan 2 a = \frac{2 \tan a}{1 - \tan^{2} a}

)

= \frac{2 \frac{12}{13}}{1 - {(\frac{12}{23})}^{2}}

(Given

\tan x = \frac{12}{13}

)

= \frac{\frac{24}{13}}{1 - \frac{144}{169}}

= \frac{\frac{24}{13}}{\frac{169 - 144}{169}}

= \frac{\frac{24}{13}}{\frac{25}{169}}

= \frac{24}{13} \times \frac{169}{25}

= \frac{24}{1} \times \frac{13}{25}

= \frac{312}{25}

= \frac{312}{25}

Hence, option 1 is correct.

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