Question

Question:-

★ Find the value of tan 9° - tan27° - tan36° + tan81°
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Answer 1

Question:

• Find the value of tan 9° - tan27° - tan36° + tan81°

Answer:

• The value of the given expression is 4

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Explanation:

Given Expression:

• tan 9° - tan 27°  - tan 36 ° + tan 81

To Find:

• The value of the expression

Formula Used

$\bigstar \; {\underline{\boxed{\bf{ Sin 2A = 2\; Sin A\; Cos A }}}}$

Required Solution:

• Simplifying the expression

$\\ \star \; {\underline{\bf{ We \; can \; write \; the\; expresion \;as :}}}$

${:\implies} \sf tan(9) + - tan(27 ) - tan(36) + tan (81)$

${:\implies} \sf tan(9 ) - tan(27 ) - cot (27) + cot (9)$

$\\ \star \; {\underline{\bf{ Writing \; tan ,cot \; terms \; into \; sin \; and \; cos :}}}$

${:\implies} \sf tan(9 ) + cot (9)- tan(27 ) - cot (27)$

${:\implies} \sf \dfrac{sin(9)}{cos(9)} - \dfrac{sin(9)}{cos(9)} - \dfrac{cos(27)}{sin(27)}+ \dfrac{cos(27)}{sin(27)}$  

$\\ \star \; {\underline{\bf{ Simplifying \; the \; expression \;further :}}}$

${:\implies} \bf \bigg(\dfrac{sin^2(9)+ cos^2 (9)}{cos(9) \times sin( 9)} \bigg)- \bigg( \dfrac{cos^2(27) + sin^2 ( 27 )}{sin(27)\times cos(27)} \bigg)$

${:\implies} \bf \dfrac{1}{cos(9) \times sin( 9)} - \dfrac{1}{sin(27)\times cos(27)}$

$\\ \star \; {\underline{\bf{ Multiplying \; numerator , denominator \; with \;2 :}}}$

${:\implies} \sf \dfrac{1}{cos(9) \times sin( 9)} - \dfrac{1}{sin(27)\times cos(27)}$

${:\implies} \sf \dfrac{2 \times1}{ 2 \times cos(9) \times sin( 9)} - \dfrac{ 2 \times 1}{2 \times sin(27)\times cos(27)}$  

$\\ \star \; {\underline{\bf{ Using \; above \; mentioned \; formula :}}}$

${:\implies} \sf \dfrac{2}{ 2 \times cos(9) \times sin( 9)} - \dfrac{ 2 }{2 \times sin(27)\times cos(27)}$

${:\implies} \sf \dfrac{2}{ sin(2 \times 9)} - \dfrac{ 2 }{sin ( 2 \times 7)}$

${:\implies} \sf \dfrac{2}{ sin( 18)} - \dfrac{ 2 }{sin( 54)}$

$\\ \star \; {\underline{\bf{ Evaluating \; the \; expression \;Further:}}}$

${:\implies} \sf 2 \bigg( \dfrac{1}{ sin( 18)} - \dfrac{ 1 }{sin( 54)} \bigg)$

${:\implies} \sf 2 \bigg( \dfrac{4}{ \sqrt{5} - 1} - \dfrac{ 4 }{\sqrt{5} + 1} \bigg)$

${:\implies} \sf 8 \bigg( \dfrac{1}{ \sqrt{5} - 1} - \dfrac{ 1 }{\sqrt{5} + 1} \bigg)$

$\\ \star \; {\underline{\bf{ Simplifying \; the \; expression \;further :}}}$

${:\implies} \sf 8 \bigg( \dfrac{\sqrt{5} + 1 - \sqrt{5} + 1}{( \sqrt{5} - 1) \sqrt{5} + 1 ) } \bigg)$

${:\implies} \sf 8 \bigg( \dfrac{1 + 1}{( \sqrt{5}) ^2 - (1 ) ^2}\bigg)$

${:\implies} \sf 8 \bigg( \dfrac{2}{4}\bigg)$

${:\implies} {\pink{\underline{\boxed{\tt{4}}}\bigstar}}$

Answer 2

Answer:

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