Question

$\large \red {\underline{ \pmb{ \bf ✿Question }}}$

find the value of :-

$\sf \bigg(1 + \dfrac{ \cos \pi }{4} \bigg) \bigg( 1 + \dfrac{ \cos3 \pi }{4} \bigg) \bigg( 1 + \dfrac{ \cos5 \pi }{4} \bigg)\bigg( 1 + \dfrac{ \cos7 \pi }{4} \bigg)$

Can we solve it by substituting the value of $\sf\dfrac{\cosπ}{4}$ in the equation ????

$\sf\pink{Class - 11th }$​

Answer 1

Step-by-step explanation:

1/4

is your answer

if you substitute the values of cos 45 , cos 135, cos 225, cos 315

which

cos 45= 1/ √2

cos135= -1/√2

cos 225= -1/√2

cos315= 1/√2

put them in ques

Answer 2

$\large{\underline{\underline{\red{\bf{Solution}}}}}$

We have four trigonometric functions. We will calculate the value of each functions separately.

$\tt\dashrightarrow{1 + \dfrac{cos \pi}{4}}$

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$\tt\dashrightarrow{1 + \dfrac{1}{\sqrt{2}}}$

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$\bf\dashrightarrow{\dfrac{\sqrt{2} + 1}{\sqrt{2}}}$⠀⠀⠀...(1)

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$\tt\dashrightarrow{1 + \dfrac{cos3 \pi}{4}}$

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$\tt\dashrightarrow{1 + \bigg( - \dfrac{1}{\sqrt{2}} \bigg)}$

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$\tt\dashrightarrow{1 - \dfrac{1}{\sqrt{2}}}$

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$\bf\dashrightarrow{\dfrac{\sqrt{2} - 1}{\sqrt{2}}}$⠀⠀⠀...(2)

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$\tt\dashrightarrow{1 + \dfrac{cos 5 \pi}{4}}$

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$\tt\dashrightarrow{1 + \bigg( - \dfrac{\sqrt{2}}{2} \bigg)}$

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$\tt\dashrightarrow{1 - \dfrac{\sqrt{2}}{2}}$

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$\bf\dashrightarrow{\dfrac{2 - \sqrt{2}}{2}}$⠀⠀⠀...(3)

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$\tt\dashrightarrow{1 + \dfrac{cos 7 \pi}{4}}$

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$\tt\dashrightarrow{1 + \dfrac{\sqrt{2}}{2}}$

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$\bf\dashrightarrow{\dfrac{2 + \sqrt{2}}{2}}$⠀⠀⠀...(4)

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We have all the values required in this question.

Now the question is, can we solve the solve the question by putting these values.

So, yes we can find the required value by putting the all values we have calculated above.

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From (1), (2), (3) and (4)

$\tt:\implies\: \: \: \: \: \: \: \: {\bigg( \dfrac{\sqrt{2} + 1}{\sqrt{2}} \bigg) \bigg(\dfrac{\sqrt{2} - 1}{\sqrt{2}} \bigg) \bigg(\dfrac{2 - \sqrt{2}}{2} \bigg)\bigg(\dfrac{2 + \sqrt{2}}{2} \bigg)}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bigg[ \dfrac{(\sqrt{2} + 1) (\sqrt{2} - 1)}{2} \bigg] \bigg[ \dfrac{(2 - \sqrt{2}) ( 2 + \sqrt{2})}{4} \bigg]}$

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Using identity

• $\sf{(a + b) (a - b) = a^2 - b^2}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bigg[ \dfrac{(\sqrt{2})^2 - (1)^2}{2} \bigg] \bigg[ \dfrac{(2)^2 - (\sqrt{2})^2}{4} \bigg]}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\bigg[ \dfrac{2 - 1}{2} \bigg] \bigg[ \dfrac{4 - 2}{4} \bigg]}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\dfrac{1}{2} \times \dfrac{2}{4}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\dfrac{1}{2} \times \dfrac{1}{2}}$

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$\frak:\implies\: \: \: \: \: \: \: \: {\underline{\boxed{\purple{\dfrac{1}{4}}}}}\: \bigstar$

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† $\small\underline{\sf{Hence,\: the\: required\: value\: is\: \dfrac{1}{4}}}$

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☆ Values used here :-

• $\sf{\dfrac{cos \pi}{4} = \dfrac{1}{\sqrt{2}}}$

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• $\sf{\dfrac{cos 3 \pi}{4} = - \dfrac{1}{\sqrt{2}}}$

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• $\sf{\dfrac{cos 5 \pi}{4} = - \dfrac{\sqrt{2}}{2}}$

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• $\sf{\dfrac{cos 7 \pi}{4} = \dfrac{\sqrt{2}}{2}}$

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