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Find the value of  \sin^8({A})+\cos^8(A)

Option: 1

1+ \frac{\sin^4(A)}{8}-\sin^2(A)


Option: 2

1+ \frac{\sin^4(2A)}{4}-\sin^2(2A)


Option: 3

1+ \frac{\sin^4(A)}{8}-\sin^2(2A)


Option: 4

1+ \frac{\sin^4(2A)}{8}-\sin^2(2A)


Answers (1)

 

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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\sin^8(A)+\cos^8(A)=(\sin^4(A)+\cos^4(A))^2-2\sin^4(A)cos^4(A)\\ =((\sin^2(A)+\cos^2(A))^2-2\sin^2(A)cos^2(A))^2 -2\sin^4(A)cos^4(A)\\ =(1-2\sin^2(A) \cos^2(A))^2-2\sin^4(A)cos^4(A)\\ =1+4 \sin^4(A) \cos^4(A)-4\sin^2(A)cos^2(A)-2\sin^4(A)cos^4(A)\\ =1+2 \sin^4(A) \cos^4(A)-4\sin^2(A)cos^2(A)\\ =1+ \frac{\sin^4(2A)}{8}-\sin^2(2A)

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Ramraj Saini

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