Question

sin 54° + cos 54°.tan 18°

​

Answer 1

Answer:

$sin54+cos54.tan18=1$

Step-by-step explanation:

Sin 54° + cos 54°.tan 18° 

$\boxed{\begin{minipage}{6cm}Formula used:\\ \\cos3A=4\:cos^3A-3\:cosA\\\\sin3A=3\:sinA-4\:sin^3A\\\\sin18^{\circ}=\frac{\sqrt{5}-1}{4}\\\\cos36^{\circ}=\frac{\sqrt{5}+1}{4}\end{minipage}}$

$sin54+cos54.tan18$

$=sin3(18)+cos3(18).\frac{sin18}{cos18}$

$=[3\:sin18-4\:sin^318]+[4\:cos^318-3\:cos18].\frac{sin18}{cos18}$

$=[3\:sin18-4\:sin^318]+cos18[4\:cos^218-3].\frac{sin18}{cos18}$

$=[3\:sin18-4\:sin^318]+[4\:cos^218-3].sin18$

$=sin18[3-4\:sin^218+4\:cos^218-3]$

$=sin18[-4\:sin^218+4\:cos^218]$

$=sin18[-4\:sin^218+4\:cos^218]$

$=4\:sin18[cos^218-sin^218]$

$=4\:sin18[\:cos2(18)]$

$=4\:sin18\:cos36$

$=4(\frac{\sqrt{5}-1}{4})(\frac{\sqrt{5}+1}{4})$

$=4(\frac{(\sqrt{5})^2-1^2}{16})$

$=4(\frac{5-1}{16})$

$=4(\frac{4}{16})$

$=1$

$\implies\:\boxed{sin54+cos54.tan18=1}$