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Question Number 205001 by rajusasmal last updated on 05/Mar/24

Answered by TonyCWX08 last updated on 05/Mar/24

$$44.$$$$x^2-y^2$$$$={(asec\left(\theta \right)+btan\left(\theta \right))}^2-{(atan\left(\theta \right)+bsec\left(\theta \right))}^2$$$$=a^2{sec}^2\left(\theta \right)+2absec\left(\theta \right)tan\left(\theta \right)+b^2{tan}^2\left(\theta \right)-a^2{tan}^2\left(\theta \right)-2abtan\left(\theta \right)sec\left(\theta \right)-b^2{sec}^2\left(\theta \right)$$$$=a^2{sec}^2\left(\theta \right)-a^2{tan}^2\left(\theta \right)+b^2{tan}^2\left(\theta \right)-b^2{sec}^2\left(\theta \right)$$$$=a^2({sec}^2\left(\theta \right)-{tan}^2\left(\theta \right))+b^2({tan}^2\left(\theta \right)-{sec}^2\left(\theta \right))$$$$=a^2\left(1\right)+b^2(-1)$$$$=a^2-b^2$$$$$$$$45.$$$$3sin\left(x\right)+5cos\left(x\right)=5$$$$3{sin}^2\left(x\right)+30sin\left(x\right)cos\left(x\right)+25{cos}^2\left(x\right)=25$$$$30sin\left(x\right)cos\left(x\right)=25-3{sin}^2\left(x\right)-25{cos}^2\left(x\right)$$$$$$$${(5sin\left(x\right)-3cos\left(x\right))}^2$$$$=25{sin}^2\left(x\right)-30sin\left(x\right)cos\left(x\right)+9{cos}^2\left(x\right)$$$$=25{sin}^2\left(x\right)-(25-3{sin}^2\left(x\right)-25{cos}^2\left(x\right))+9{cos}^2\left(x\right)$$$$=25{sin}^2\left(x\right)-25+3{sin}^2\left(x\right)+25{cos}^2\left(x\right)+9{cos}^2\left(x\right)$$$$=34{sin}^2\left(x\right)+34{cos}^2\left(x\right)-25$$$$=34({sin}^2\left(x\right)+{cos}^2\left(x\right))-25$$$$=34-25$$$$=9$$$$$$$$5sin\left(x\right)-3cos\left(x\right)=\pm \sqrt{9}=\pm 3$$$$$$

Answered by TonyCWX08 last updated on 05/Mar/24

$$36.$$$$x^2+y^2+z^2$$$$=r^2{sin}^2\left(A\right){cos}^2\left(C\right)+r^2{sin}^2\left(A\right){sin}^2\left(C\right)+r^2{cos}^2\left(A\right)$$$$=r^2({sin}^2\left(A\right)({sin}^2\left(C\right)+{cos}^2\left(C\right))+{cos}^2\left(A\right))$$$$=r^2({sin}^2\left(A\right)+{cos}^2\left(A\right))$$$$=r^2\left(1\right)$$$$=r^2$$$$$$$$40.$$$$sin\left(x\right)+{sin}^2\left(x\right)=1$$$${sin}^2\left(x\right)=1-sin\left(x\right)$$$$1-{cos}^2\left(x\right)=1-sin\left(x\right)$$$$-{cos}^2\left(x\right)=-sin\left(x\right)$$$${cos}^2\left(x\right)=sin\left(x\right)$$$${cos}^4\left(x\right)={sin}^2\left(x\right)$$$${cos}^2\left(x\right)+{cos}^4\left(x\right)$$$$=sin\left(x\right)+{sin}^2\left(x\right)=1$$

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