Question

The value of the expression log10(tan6°) + log10(tan12°) + log10(tan18°) + ...+ log10(tan84°) is

Answer 1

$\displaystyle \sf log_{10}(tan {6}^{ \circ} ) + log_{10}(tan {12}^{ \circ} ) + log_{10}(tan {18}^{ \circ} ) + ... + log_{10}(tan {84}^{ \circ} ) = \bf 0$

Given :

$\displaystyle \sf log_{10}(tan {6}^{ \circ} ) + log_{10}(tan {12}^{ \circ} ) + log_{10}(tan {18}^{ \circ} ) + ... + log_{10}(tan {84}^{ \circ} )$

To find :

The value of the expression

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf log_{10}(tan {6}^{ \circ} ) + log_{10}(tan {12}^{ \circ} ) + log_{10}(tan {18}^{ \circ} ) + ... + log_{10}(tan {84}^{ \circ} )$

Step 2 of 2 :

Find the value of the expression

$\displaystyle \sf log_{10}(tan {6}^{ \circ} ) + log_{10}(tan {12}^{ \circ} ) + log_{10}(tan {18}^{ \circ} ) + ... + log_{10}(tan {84}^{ \circ} )$

Rearranging the terms we get

$\displaystyle \sf = log_{10}(tan {6}^{ \circ} ) + log_{10}(tan {84}^{ \circ} ) + log_{10}(tan {12}^{ \circ} ) + log_{10}(tan {78}^{ \circ} ) + log_{10}(tan {18}^{ \circ} ) + log_{10}(tan {72}^{ \circ} ) + ... + log_{10}(tan {42}^{ \circ} ) + log_{10}(tan {48}^{ \circ} )$

$\displaystyle \sf = log_{10}(tan {6}^{ \circ} \times tan {84}^{ \circ} ) + log_{10}(tan {12}^{ \circ} \times tan {78}^{ \circ} ) + log_{10}(tan {18}^{ \circ} \times tan {72}^{ \circ} ) + ... + log_{10}(tan {42}^{ \circ} \times tan {48}^{ \circ} ) \: \: \: \bigg[ \: \because \: log(a) + log(b) = log(ab) \bigg]$

$\displaystyle \sf = log_{10}(tan {6}^{ \circ} \times tan [{90}^{ \circ} -{6}^{ \circ} ]) + log_{10}(tan {12}^{ \circ} \times tan [{90}^{ \circ} -{12}^{ \circ} ]) + log_{10}(tan {18}^{ \circ} \times tan [{90}^{ \circ} -{18}^{ \circ} ])+ ... + log_{10}(tan {42}^{ \circ} \times tan [{90}^{ \circ} -{42}^{ \circ} ])$

$\displaystyle \sf = log_{10}( tan {6}^{ \circ} \times cot {6}^{ \circ}) +log_{10}( tan {12}^{ \circ} \times cot {12}^{ \circ}) + log_{10}( tan {18}^{ \circ} \times cot {18}^{ \circ}) + ... + log_{10}( tan {42}^{ \circ} \times cot {42}^{ \circ})$

$\displaystyle \sf = log_{10}(1 ) + log_{10}(1 ) + log_{10}(1 ) + ... + log_{10}(1 ) \: \: \: \bigg[ \: \because \: tan \theta\times cot \theta = 1\bigg]$

$= 0 + 0 + 0 + ... + 0$

$= 0$

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$ If 2log((x+y)/(4))=log x+log y then find the value of (x)/(y)+(y)/(x)

https://brainly.in/question/24081206

2. If log (0.0007392)=-3.1313 , then find log(73.92)

https://brainly.in/question/21121422

#SPJ3