MHT CET · Maths · Trigonometric Ratios & Identities
\(\tan A+2 \tan 2 A+4 \tan 4 A+8 \cot 8 A=\)
- A \(\tan 2 \mathrm{~A}\)
- B \(\cot \mathrm{A}\)
- C \(\tan \mathrm{A}\)
- D \(\cot 2 \mathrm{~A}\)
Answer & Solution
Correct Answer
(B) \(\cot \mathrm{A}\)
Step-by-step Solution
Detailed explanation
\(\tan A+2 \tan 2 A+4 \tan 4 A+8 \cot 8 A \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+\frac{8}{\left(\frac{2 \tan 4 \mathrm{~A}}{1-\tan ^2 4 \mathrm{~A}}\right)} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+\frac{8\left(1-\tan ^2 \mathrm{~A}\right)}{2 \tan 4 \mathrm{~A}} \)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\) \(\frac{(4 \tan 4 \mathrm{~A})(2 \tan 4 \mathrm{~A})+8\left(1-\tan ^2 4 \mathrm{~A}\right)}{2 \tan 4 \mathrm{~A}} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8}{2 \tan 4 \mathrm{~A}} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8}{2\left(\frac{2 \tan \mathrm{A}}{1-\tan ^2 2 \mathrm{~A}}\right)} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8\left(1-\tan ^2 2 \mathrm{~A}\right)}{4 \tan 2 \mathrm{~A}} \)
\( =\tan \mathrm{A}+\frac{(2 \tan 2 \mathrm{~A})(4 \tan 2 \mathrm{~A})+8\left(1-\tan ^2 2 \mathrm{~A}\right)}{4 \tan 2 \mathrm{~A}}\) \(=\tan \mathrm{A}+\frac{8}{4 \tan 2 \mathrm{~A}} \)
\( =\tan \mathrm{A}+\frac{8}{4\left(\frac{2 \tan \mathrm{A}}{1-\tan ^2 \mathrm{~A}}\right)}=\tan \mathrm{A}+\frac{8\left(1-\tan ^2 \mathrm{~A}\right)}{8 \tan \mathrm{A}}\)
\(=\frac{\tan \mathrm{A}(8 \tan \mathrm{A})+8\left(1-\tan ^2 \mathrm{~A}\right)}{8 \tan \mathrm{A}}=\frac{8}{8 \tan \mathrm{A}}=\cot \mathrm{A}\)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+\frac{8}{\left(\frac{2 \tan 4 \mathrm{~A}}{1-\tan ^2 4 \mathrm{~A}}\right)} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+\frac{8\left(1-\tan ^2 \mathrm{~A}\right)}{2 \tan 4 \mathrm{~A}} \)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\) \(\frac{(4 \tan 4 \mathrm{~A})(2 \tan 4 \mathrm{~A})+8\left(1-\tan ^2 4 \mathrm{~A}\right)}{2 \tan 4 \mathrm{~A}} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8}{2 \tan 4 \mathrm{~A}} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8}{2\left(\frac{2 \tan \mathrm{A}}{1-\tan ^2 2 \mathrm{~A}}\right)} \)
\( =\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{8\left(1-\tan ^2 2 \mathrm{~A}\right)}{4 \tan 2 \mathrm{~A}} \)
\( =\tan \mathrm{A}+\frac{(2 \tan 2 \mathrm{~A})(4 \tan 2 \mathrm{~A})+8\left(1-\tan ^2 2 \mathrm{~A}\right)}{4 \tan 2 \mathrm{~A}}\) \(=\tan \mathrm{A}+\frac{8}{4 \tan 2 \mathrm{~A}} \)
\( =\tan \mathrm{A}+\frac{8}{4\left(\frac{2 \tan \mathrm{A}}{1-\tan ^2 \mathrm{~A}}\right)}=\tan \mathrm{A}+\frac{8\left(1-\tan ^2 \mathrm{~A}\right)}{8 \tan \mathrm{A}}\)
\(=\frac{\tan \mathrm{A}(8 \tan \mathrm{A})+8\left(1-\tan ^2 \mathrm{~A}\right)}{8 \tan \mathrm{A}}=\frac{8}{8 \tan \mathrm{A}}=\cot \mathrm{A}\)
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