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  • The value of  <img alt="\cos ^{3}\left ( \frac{\pi }{8} \right )\cdot \cos \left ( \frac{3\pi }{8} \right )+\sin ^{3}\left ( \frac{\pi }{8} \right )\cdot \sin \left ( \frac{3\pi }{8} \right )" src="https://learn.careers360.com/latex-image/?%5Ccos%

The value of  \cos ^{3}\left ( \frac{\pi }{8} \right )\cdot \cos \left ( \frac{3\pi }{8} \right )+\sin ^{3}\left ( \frac{\pi }{8} \right )\cdot \sin \left ( \frac{3\pi }{8} \right )  is :
Option: 1 \frac{1}{4}
 
Option: 2 \frac{1}{2\sqrt{2}}
 
Option: 3 \frac{1}{2}
 
Option: 4 \frac{1}{\sqrt{2}}
 
 

Answers (2)

best_answer

 

 

Trigonometric Identities -

Trigonometric Identities-

These identities are the equations that hold true regardless of the angle being chosen.

 

\\\mathrm{\sin^2\mathit{t}+\cos^2\mathit{t}=1}\\\mathrm{1+\tan^2\mathit{t}=\sec^2\mathit{t}}\\\mathrm1+{\cot^2\mathit{t}=\csc^2\mathit{t}}\\\mathrm{\tan \mathit{t}=\frac{\sin \mathit{t}}{\cos \mathit{t}},\;\;\cot \mathit{t}=\frac{\cos\mathit{t}}{\sin\mathit{t}}}

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Allied Angles (Part 1) -

Allied Angles (Part 1)

Two angles or numbers are called allied iff their sum or difference is a multiple of π/2   

  • sin (900 - θ) = cos (θ)

  • cos (900 - θ) = sin (θ)

  • tan (900 - θ) = cot (θ)

  • csc (900 - θ) = sec (θ)          

  • sec (900 - θ) = csc (θ)

  • cot (900 - θ) = tan (θ)

 

  • sin (900 + θ) = cos (θ)

  • cos (900 + θ) = - sin (θ)

  • tan (900 + θ) = - cot (θ)

  • csc (900 + θ) = sec (θ)          

  • sec (900 + θ) = - csc (θ)

  • cot (900 + θ) = - tan (θ)

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\\\cos ^3\:\frac{\pi }{8}\cos \frac{3\:\pi }{8}+\sin ^3\frac{\pi }{8}\:\sin \frac{3\:\pi }{8}\\\sin\left ( \frac{\pi}{2}-\frac{3\pi}{8} \right )=\cos\left ( \frac{3\pi}{8} \right )=\sin\left ( \frac{\pi}{8} \right )\\\\\\\cos ^3\:\frac{\pi }{8}\sin \frac{\:\pi }{8}+\sin ^3\frac{\pi }{8}\:\cos \frac{\:\pi }{8}\\\sin\frac{\pi}{8}\cos\frac{\pi}{8}\left ( \cos ^2\:\frac{\pi }{8}+\sin ^2\:\frac{\pi }{8} \right )\\\frac{1}{2}\left ( 2\:\sin \frac{\pi }{8}\cos \:\frac{\pi \:}{8} \right )=\frac{1}{2}\:\sin \frac{2\pi }{8}=\frac{1}{2\sqrt2}

Posted by

avinash.dongre

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JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Option 2

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sangeeta biradar

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