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Popular Trigonometry Problems

prove (1+csc(x))(1-sin(x))=cot(x)cos(x)	prove\:(1+\csc(x))(1-\sin(x))=\cot(x)\cos(x)
prove arcsec(x)= 1/(sec(x))	prove\:\arcsec(x)=\frac{1}{\sec(x)}
prove cos(4x)cos(2x)=cos^2(3x)-cos^2(x)	prove\:\cos(4x)\cos(2x)=\cos^{2}(3x)-\cos^{2}(x)
prove cos(4x)-sin(4x)=1-2sin(2x)	prove\:\cos(4x)-\sin(4x)=1-2\sin(2x)
prove cos(x)tan^3(x)=sin(x)tan^2(x)	prove\:\cos(x)\tan^{3}(x)=\sin(x)\tan^{2}(x)
prove sin^2(0)=4(cos(-0)-1)	prove\:\sin^{2}(0)=4(\cos(-0)-1)
prove sec(2a)+tan(2a)=(cos(a)+sin(a))/(cos(a)-sin(a))	prove\:\sec(2a)+\tan(2a)=\frac{\cos(a)+\sin(a)}{\cos(a)-\sin(a)}
prove sec^2(x)= 1/(1-sin^2(x))	prove\:\sec^{2}(x)=\frac{1}{1-\sin^{2}(x)}
prove cos(-x)+sin(-x)=cos(x)-sin(x)	prove\:\cos(-x)+\sin(-x)=\cos(x)-\sin(x)
prove (1-cos(2θ))/(cos(2θ)+1)=tan^2(θ)	prove\:\frac{1-\cos(2θ)}{\cos(2θ)+1}=\tan^{2}(θ)
prove cos(x-pi/3)-sin(x+pi/6)=0	prove\:\cos(x-\frac{π}{3})-\sin(x+\frac{π}{6})=0
prove ((cos(x)+1))/(sin^3(x))=((csc(x)))/(1-cos(x))	prove\:\frac{(\cos(x)+1)}{\sin^{3}(x)}=\frac{(\csc(x))}{1-\cos(x)}
prove (1+cot^2(x))/(sec^2(x))=cot^2(x)	prove\:\frac{1+\cot^{2}(x)}{\sec^{2}(x)}=\cot^{2}(x)
prove 2cos^2(x/2)=(sin^2(x))/(1-cos(x))	prove\:2\cos^{2}(\frac{x}{2})=\frac{\sin^{2}(x)}{1-\cos(x)}
prove sin(a)cos(a)=cos^2(a)tan(a)	prove\:\sin(a)\cos(a)=\cos^{2}(a)\tan(a)
prove sin(-pi/4)=-sin(pi/4)	prove\:\sin(-\frac{π}{4})=-\sin(\frac{π}{4})
prove sin(z+w)=sin(z)cos(w)+cos(z)sin(w)	prove\:\sin(z+w)=\sin(z)\cos(w)+\cos(z)\sin(w)
prove tan^2(x)=((1-cos(2x)))/(1+cos(2x))	prove\:\tan^{2}(x)=\frac{(1-\cos(2x))}{1+\cos(2x)}
prove cos(0)tan(0)=sin(0)	prove\:\cos(0)\tan(0)=\sin(0)
prove cos(pi/6)= 4/3 cos^3(pi/6)	prove\:\cos(\frac{π}{6})=\frac{4}{3}\cos^{3}(\frac{π}{6})
prove tan(-pi/4)=-1	prove\:\tan(-\frac{π}{4})=-1
prove sin^2(x)= 1/(1+cot^2(x))	prove\:\sin^{2}(x)=\frac{1}{1+\cot^{2}(x)}
prove (2sec^2(x)-2tan^2(x))/(csc(x))=sin(2x)sec(x)	prove\:\frac{2\sec^{2}(x)-2\tan^{2}(x)}{\csc(x)}=\sin(2x)\sec(x)
prove sin(2α)=2sin(α)cos(α)	prove\:\sin(2α)=2\sin(α)\cos(α)
prove (1+csc(x))(1+csc(-x))=-cot^2(x)	prove\:(1+\csc(x))(1+\csc(-x))=-\cot^{2}(x)
prove cos(0)-sec(0)=-sin(0)tan(0)	prove\:\cos(0)-\sec(0)=-\sin(0)\tan(0)
prove tan(pi/6)= 1/(sqrt(3))	prove\:\tan(\frac{π}{6})=\frac{1}{\sqrt{3}}
prove cos(4a)=1-8sin^2(a)cos^2(a)	prove\:\cos(4a)=1-8\sin^{2}(a)\cos^{2}(a)
prove 1-cos(4x)=2sin^2(2x)	prove\:1-\cos(4x)=2\sin^{2}(2x)
prove cos(2β)=(1-tan^2(β))/(1+tan^2(β))	prove\:\cos(2β)=\frac{1-\tan^{2}(β)}{1+\tan^{2}(β)}
prove tan(pi+x)=tan(x)	prove\:\tan(π+x)=\tan(x)
prove tan(A)=tan(A)csc^2(A)+cot(-A)	prove\:\tan(A)=\tan(A)\csc^{2}(A)+\cot(-A)
prove 1/(1+sin(x))=(sec(x)-tan(x))sec(x)	prove\:\frac{1}{1+\sin(x)}=(\sec(x)-\tan(x))\sec(x)
prove tan(a)+cot(a)= 2/(sin(2a))	prove\:\tan(a)+\cot(a)=\frac{2}{\sin(2a)}
prove (1+sin(α))/(cos(α))=(cos(α))/(1-sin(α))	prove\:\frac{1+\sin(α)}{\cos(α)}=\frac{\cos(α)}{1-\sin(α)}
prove 2csc^2(y)= 1/(1-cos(y))+1/(1+cos(y))	prove\:2\csc^{2}(y)=\frac{1}{1-\cos(y)}+\frac{1}{1+\cos(y)}
prove cot(2x)=(cos(2x))/(sin(2x))	prove\:\cot(2x)=\frac{\cos(2x)}{\sin(2x)}
prove sin(θ-pi/2)=-cos(θ)	prove\:\sin(θ-\frac{π}{2})=-\cos(θ)
prove sec(x)-cos(x)=tan(x)*sin(x)	prove\:\sec(x)-\cos(x)=\tan(x)\cdot\:\sin(x)
prove cot(45)=(cos(45))/(sin(45))	prove\:\cot(45^{\circ\:})=\frac{\cos(45^{\circ\:})}{\sin(45^{\circ\:})}
prove sqrt((1-sin(x))/(1+sin(x)))=sec(x)-tan(x)	prove\:\sqrt{\frac{1-\sin(x)}{1+\sin(x)}}=\sec(x)-\tan(x)
prove cot(2x)+csc(2x)=cot(x)	prove\:\cot(2x)+\csc(2x)=\cot(x)
prove sin(x-pi/2)+sin(x+pi/2)=0	prove\:\sin(x-\frac{π}{2})+\sin(x+\frac{π}{2})=0
prove sin(-x)+csc(x)=cot(x)*cos(x)	prove\:\sin(-x)+\csc(x)=\cot(x)\cdot\:\cos(x)
prove tan(45+x)+tan(45-x)=2sec(2x)	prove\:\tan(45^{\circ\:}+x)+\tan(45^{\circ\:}-x)=2\sec(2x)
prove cos^2(8x)-sin^2(8x)=cos(16x)	prove\:\cos^{2}(8x)-\sin^{2}(8x)=\cos(16x)
prove cos^2(2x)= 1/2+1/2 cos(4x)	prove\:\cos^{2}(2x)=\frac{1}{2}+\frac{1}{2}\cos(4x)
prove 2sin(-t)cos(-t)=2sin(-t)cos(t)	prove\:2\sin(-t)\cos(-t)=2\sin(-t)\cos(t)
prove sin^2(α)+sin^2(α)tan^2(α)=tan^2(α)	prove\:\sin^{2}(α)+\sin^{2}(α)\tan^{2}(α)=\tan^{2}(α)
prove cos(75)=sqrt((1+cos(150))/2)	prove\:\cos(75^{\circ\:})=\sqrt{\frac{1+\cos(150^{\circ\:})}{2}}
prove cos(7x)-cos(3x)=-2sin(5x)sin(2x)	prove\:\cos(7x)-\cos(3x)=-2\sin(5x)\sin(2x)
prove (sec^2(a))/(2-sec^2(a))=sec(2a)	prove\:\frac{\sec^{2}(a)}{2-\sec^{2}(a)}=\sec(2a)
prove cot(a)+tan(a)=csc(a)sec(a)	prove\:\cot(a)+\tan(a)=\csc(a)\sec(a)
prove sin(a+b)=sin(a)+sin(b)	prove\:\sin(a+b)=\sin(a)+\sin(b)
prove sin(120)=(sqrt(3))/2	prove\:\sin(120^{\circ\:})=\frac{\sqrt{3}}{2}
prove sin((2pi)/3)=sin(pi/3)	prove\:\sin(\frac{2π}{3})=\sin(\frac{π}{3})
prove cos(x)(csc^2(x))=cot(x)csc(x)	prove\:\cos(x)(\csc^{2}(x))=\cot(x)\csc(x)
prove sin(x)cos(x)csc^2(x)=cot(x)	prove\:\sin(x)\cos(x)\csc^{2}(x)=\cot(x)
prove 2-2tan(x)cot(2x)=sec^2(x)	prove\:2-2\tan(x)\cot(2x)=\sec^{2}(x)
prove sec^2(B)-csc^2(B)=(tan(B)-cot(B))/(sin(B)cos(B))	prove\:\sec^{2}(B)-\csc^{2}(B)=\frac{\tan(B)-\cot(B)}{\sin(B)\cos(B)}
prove 2+sec(a)cos(a)=3	prove\:2+\sec(a)\cos(a)=3
prove 5cos^2(θ)+2sin^2(θ)=3cos^2(θ)+2	prove\:5\cos^{2}(θ)+2\sin^{2}(θ)=3\cos^{2}(θ)+2
prove (sec(a))/(tan(a)+cot(a))=sin(a)	prove\:\frac{\sec(a)}{\tan(a)+\cot(a)}=\sin(a)
prove sin((9pi)/2+x)=cos(x)	prove\:\sin(\frac{9π}{2}+x)=\cos(x)
prove cos^4(β)-sin^4(β)=cos(2β)	prove\:\cos^{4}(β)-\sin^{4}(β)=\cos(2β)
prove cos^3(x)= 3/4 cos(x)+1/4 cos(3x)	prove\:\cos^{3}(x)=\frac{3}{4}\cos(x)+\frac{1}{4}\cos(3x)
prove tan(pi/4+x)=(1+sin(2x))/(cos(2x))	prove\:\tan(\frac{π}{4}+x)=\frac{1+\sin(2x)}{\cos(2x)}
prove sin^2(x)+1/(sec^2(x))=sin(x)csc(x)	prove\:\sin^{2}(x)+\frac{1}{\sec^{2}(x)}=\sin(x)\csc(x)
prove sin(x)=sqrt(1-cos(x))	prove\:\sin(x)=\sqrt{1-\cos(x)}
prove (sec(2x)+1)/(tan(2x))=cot(x)	prove\:\frac{\sec(2x)+1}{\tan(2x)}=\cot(x)
prove cos^2(6θ)-sin^2(6θ)=cos(12θ)	prove\:\cos^{2}(6θ)-\sin^{2}(6θ)=\cos(12θ)
prove (cos(β))/(tan(β))=csc(β)-sin(β)	prove\:\frac{\cos(β)}{\tan(β)}=\csc(β)-\sin(β)
prove cos(3t)=4cos^3(t)-3cos(t)	prove\:\cos(3t)=4\cos^{3}(t)-3\cos(t)
prove sec(2x)=(sec^2(x)+sec^4(x))/(2+sec^2(x)-sec^4(x))	prove\:\sec(2x)=\frac{\sec^{2}(x)+\sec^{4}(x)}{2+\sec^{2}(x)-\sec^{4}(x)}
prove 1/(csc(x))= 1/(sin(x))	prove\:\frac{1}{\csc(x)}=\frac{1}{\sin(x)}
prove 1+tan^2(A)=sec^3(A)cos(A)	prove\:1+\tan^{2}(A)=\sec^{3}(A)\cos(A)
prove sec(0)-cos(0)=tan(0)sin(0)	prove\:\sec(0)-\cos(0)=\tan(0)\sin(0)
prove 1-sin(θ)cos(θ)tan(θ)=cos^2(θ)	prove\:1-\sin(θ)\cos(θ)\tan(θ)=\cos^{2}(θ)
prove tan^2(x)+sin(x)csc(x)=sec^2(x)	prove\:\tan^{2}(x)+\sin(x)\csc(x)=\sec^{2}(x)
prove cos^2(9θ)-sin^2(9θ)=cos(18θ)	prove\:\cos^{2}(9θ)-\sin^{2}(9θ)=\cos(18θ)
prove sec(x)csc(x)=2csc(2x)	prove\:\sec(x)\csc(x)=2\csc(2x)
prove sec(-θ)=sec(θ)	prove\:\sec(-θ)=\sec(θ)
prove sin((11pi}{12})=sin(\frac{3pi)/4+pi/6)	prove\:\sin(\frac{11π}{12})=\sin(\frac{3π}{4}+\frac{π}{6})
prove (sin(2x))/(1-sin^2(x))=2tan(x)	prove\:\frac{\sin(2x)}{1-\sin^{2}(x)}=2\tan(x)
prove sin(4x)=-2sin(2x)	prove\:\sin(4x)=-2\sin(2x)
prove sin(2t)=2sin(t)cos(t)	prove\:\sin(2t)=2\sin(t)\cos(t)
prove (csc(x))/(2cos(x))=csc(2x)	prove\:\frac{\csc(x)}{2\cos(x)}=\csc(2x)
prove tan^2(u)=(1-cos(2u))/(1+cos(2u))	prove\:\tan^{2}(u)=\frac{1-\cos(2u)}{1+\cos(2u)}
prove csc^4(x)-csc^2(x)=cot^2(+cot^4(x))	prove\:\csc^{4}(x)-\csc^{2}(x)=\cot^{2}(+\cot^{4}(x))
prove (cot^2(x))/(csc(x)+1)+1=csc(x)	prove\:\frac{\cot^{2}(x)}{\csc(x)+1}+1=\csc(x)
prove (sin^2(θ))/(1-cos(θ))-1=cos(θ)	prove\:\frac{\sin^{2}(θ)}{1-\cos(θ)}-1=\cos(θ)
prove 1-sin(y)=(cos^2(y))/(1+sin(y))	prove\:1-\sin(y)=\frac{\cos^{2}(y)}{1+\sin(y)}
prove 2tan(x)=(2sin(x))/(cos(x))	prove\:2\tan(x)=\frac{2\sin(x)}{\cos(x)}
prove 1/(1-sin(B))=sec^2(B)+sec(B)tan(B)	prove\:\frac{1}{1-\sin(B)}=\sec^{2}(B)+\sec(B)\tan(B)
prove tan(60)=(2tan(30))/(1-tan^2(30))	prove\:\tan(60^{\circ\:})=\frac{2\tan(30^{\circ\:})}{1-\tan^{2}(30^{\circ\:})}
prove sin^2(u)(cot^2(u)+1)=1	prove\:\sin^{2}(u)(\cot^{2}(u)+1)=1
prove (cos(-x))/(sin(-x))=-cot(x)	prove\:\frac{\cos(-x)}{\sin(-x)}=-\cot(x)
prove sin((9pi)/4)=cos((9pi)/4)	prove\:\sin(\frac{9π}{4})=\cos(\frac{9π}{4})
prove tan(β)=csc(β)sec(β)-cot(β)	prove\:\tan(β)=\csc(β)\sec(β)-\cot(β)
prove cos(x-90)=sin(x)	prove\:\cos(x-90^{\circ\:})=\sin(x)

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