Question

If 2cos theta - sin theta =0 then tan4 theta + cot⁴ theta is
$\frac{257}{16}$
True or false? ​

Answer 1

True

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Answer 2

$\tan^4 \theta$ + $\cot^4 \theta$ = $\dfrac{257}{16}$, true.

Step-by-step explanation:

We have,

2$\cos \theta$ - $\sin \theta$ = 0

State $\tan^4 \theta$ + $\cot^4 \theta$ = $\dfrac{257}{16}$, True or False.

∴ 2$\cos \theta$ - $\sin \theta$ = 0

⇒ $\sin \theta$ = 2$\cos \theta$

⇒ $\dfrac{\sin \theta}{\cos \theta}$ = 2

⇒ $\tan \theta$ = 2

⇒ $\tan^4 \theta$ = $2^4$ = 16 [ ∵ $\cot A = \dfrac{1}{\tan A}$]

∴ $\cot^4 \theta$ = $\dfrac{1}{16}$

L.H.S. = $\tan^4 \theta$ + $\cot^4 \theta$

= 16 + $\dfrac{1}{16}$

= $\dfrac{256+1}{16}$

= $\dfrac{257}{16}$

= R.H.S., True.

Thus, $\tan^4 \theta$ + $\cot^4 \theta$ = $\dfrac{257}{16}$, true.