Question

cos36-cos72 I need clear step by step explanation ​

Answer 1

Answer:

1/2 is the answer and attachment is there

Answer 2

Answer:

Question :-

• Evaluate cos36° - cos72°

Answer

Concept used :-

Transformation Formula :-

$\bf \:cosC - cosD = - 2sin(\dfrac{C + D}{2} )sin(\dfrac{C - D}{2} )$

$\bf \:sin18° \: = \dfrac{ \sqrt{5} - 1 }{4}$

$\bf \:sin54° \: = \dfrac{ \sqrt{5} + 1 }{4}$

Solution :-

Consider cos36° - cos72°

Using identity, we get

$\bf\implies \: - 2sin(\dfrac{ 36° + 72°}{2} )sin(\dfrac{36° - 72°}{2} )$

$\bf\implies \: - 2sin54°sin18°$

$\bf\implies \:2sin18°sin54°$

On substituting the values of sin18° and sin54°, we get

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

$\bf\implies \:2 \times (\dfrac{5 - 1}{16} )$

$\bf\implies \:\dfrac{1}{2}$

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Other important values :-

$\bf \:sin18° = cos72°= \dfrac{ \sqrt{5} - 1}{4}$

$\bf \:sin36° = cos54° = \dfrac{ \sqrt{10 - 2 \sqrt{5} } }{4}$

$\bf \:sin54° = cos36° = \dfrac{ \sqrt{5} + 1}{4}$

$\bf \:sin72° = cos18° = \dfrac{ \sqrt{10 + 2 \sqrt{5} } }{4}$