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| 1. |
Prove that Cos^4A-Sin^4A+1=2Cos^2A |
| Answer» <html><body><p><strong>Answer:</strong></p><p>I <a href="https://interviewquestions.tuteehub.com/tag/learned-536887" style="font-weight:bold;" target="_blank" title="Click to know more about LEARNED">LEARNED</a> this from a friend. Hope it helps</p><p><strong>Step-by-step explanation:</strong></p><p>First, group the first two terms and factor it.</p><p></p><p>(cos^4A-sin^4A)+<a href="https://interviewquestions.tuteehub.com/tag/1-256655" style="font-weight:bold;" target="_blank" title="Click to know more about 1">1</a>=2cos^2A</p><p>(cos^2A-sin^2A)(cos^2A+sin^2A)+1=2cos^2A</p><p>Then, apply the Pythagorean identity which is cos^2theta + sin^2theta=1 .</p><p></p><p> </p><p></p><p>(cos^2A-sin^2A)(1)+1=2cos^2A</p><p>cos^2A-sin^2A + 1=2cos^2A</p><p>Then, group the second and last <a href="https://interviewquestions.tuteehub.com/tag/term-1241851" style="font-weight:bold;" target="_blank" title="Click to know more about TERM">TERM</a> at the left side of the equation.</p><p></p><p>cos^2A+(-sin^2A+1)=2cos^2A</p><p>cos^2A+(1-sin^2A)=2cos^2A</p><p>To <a href="https://interviewquestions.tuteehub.com/tag/simplify-644378" style="font-weight:bold;" target="_blank" title="Click to know more about SIMPLIFY">SIMPLIFY</a> the expression <a href="https://interviewquestions.tuteehub.com/tag/inside-1045864" style="font-weight:bold;" target="_blank" title="Click to know more about INSIDE">INSIDE</a> the parenthesis, apply the Pythagorean identity again.</p><p></p><p>cos^2A+cos^2A=2cos^2A</p><p>2cos^2A=2cos^2A</p><p>Since left side simplifies to 2cos^2A which is the same term with the right side, hence it proves that the given equation is an identity.</p></body></html> | |