| Summary: | Using the power series expansions of the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">cot</mo><mi>x</mi><mo>,</mo><mn>1</mn><mo>/</mo><mo form="prefix">sin</mo><mi>x</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mo form="prefix">sin</mo><mn>2</mn></msup><mi>x</mi></mrow></semantics></math></inline-formula>, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>π</mi><mo>/</mo><mn>2</mn></mfenced></semantics></math></inline-formula>. Our results are much better than those in the existing literature.
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