Question

$\cos ecA-2cot2A\cos A=?$

• A. $2\sin A$
• B. $secA$
• C. $2\cos AcotA$
• D. None of these

Answer 1

The correct option is A

$2\sin A$

Explanation for the correct option:

Step 1. Find the value of $\cos ecA–2cot2A\cos A$

$=\cos ecA–\left(\frac{2\cos A}{\tan 2A}\right)$

Step 2. Use the identity $\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathbf{2}\mathit{x}\mathbf{}\mathbf{=}\frac{\mathbf{}\mathbf{2}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{x}}{\mathbf{1}\mathbf{}\mathbf{–}\mathbf{}\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{2}}\mathbf{x}}$,

$=\cos ecA–\left(\frac{2\cos A}{2\tan A}\right)(1–{\tan }^{2}A)$

$=\cos ecA–\left[\frac{\cos A}{\left({\displaystyle \frac{\sin A}{\cos A}}\right)}\right]\left[1-\frac{{\sin }^{2}A}{{\cos }^{2}A}\right]$

$=\cos ecA–[\cos ecA{\cos }^{2}A]\left[\frac{({\cos }^{2}A–{\sin }^{2}A)}{{\cos }^{2}A}\right]$

$=\cos ecA–\cos ecA({\cos }^{2}A–{\sin }^{2}A)$

$=\cos ecA[1–{\cos }^{2}A+{\sin }^{2}A]$

Step 4. Using the identity $\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}$,

$=\cos ecA[{\sin }^{2}A+{\sin }^{2}A]$

$=\frac{1}{\sin A}\times 2{\sin }^{2}A$

$=\mathbf{2}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{A}$

Hence, Option ‘A' is Correct.