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Popular Trigonometry Problems

prove csc(2x)=(csc(x))/(2cos(x))	prove\:\csc(2x)=\frac{\csc(x)}{2\cos(x)}
prove (1-tan(x))/(1+tan(x))=(1-sin(2x))/(cos(2x))	prove\:\frac{1-\tan(x)}{1+\tan(x)}=\frac{1-\sin(2x)}{\cos(2x)}
prove cos(θ+pi/2)=-sin(θ)	prove\:\cos(θ+\frac{π}{2})=-\sin(θ)
prove sin^2(a)+cos^2(a)=1	prove\:\sin^{2}(a)+\cos^{2}(a)=1
prove tan(x/2)+cos(x)tan(x/2)=sin(x)	prove\:\tan(\frac{x}{2})+\cos(x)\tan(\frac{x}{2})=\sin(x)
prove csch^2(x)=coth^2(x)-1	prove\:\csch^{2}(x)=\coth^{2}(x)-1
prove 1/(sec(x)+tan(x))=sec(x)-tan(x)	prove\:\frac{1}{\sec(x)+\tan(x)}=\sec(x)-\tan(x)
prove (tan(y))/(csc(y))= 1/(cos(y))-1/(sec(y))	prove\:\frac{\tan(y)}{\csc(y)}=\frac{1}{\cos(y)}-\frac{1}{\sec(y)}
prove (1+sin(x))/(cos(x))=sec(x)+tan(x)	prove\:\frac{1+\sin(x)}{\cos(x)}=\sec(x)+\tan(x)
prove 2tan(x)sec(x)= 1/(1-sin(x))-1/(1+sin(x))	prove\:2\tan(x)\sec(x)=\frac{1}{1-\sin(x)}-\frac{1}{1+\sin(x)}
prove sin^2(x)(csc^2(x)-1)=cos^2(x)	prove\:\sin^{2}(x)(\csc^{2}(x)-1)=\cos^{2}(x)
prove 1/(sin(x)cot(x))= 1/(cos(x))	prove\:\frac{1}{\sin(x)\cot(x)}=\frac{1}{\cos(x)}
prove (csc^2(x)-1)/(csc^2(x))=cos^2(x)	prove\:\frac{\csc^{2}(x)-1}{\csc^{2}(x)}=\cos^{2}(x)
prove (sin(x/2)+cos(x/2))^2=1+sin(x)	prove\:(\sin(\frac{x}{2})+\cos(\frac{x}{2}))^{2}=1+\sin(x)
prove cos(3α)=4cos^3(α)-3cos(α)	prove\:\cos(3α)=4\cos^{3}(α)-3\cos(α)
prove 1/(sin(2θ))-1/(tan(2θ))=tan(θ)	prove\:\frac{1}{\sin(2θ)}-\frac{1}{\tan(2θ)}=\tan(θ)
prove sin(x)cot(x)+cos(x)tan^2(x)=sec(x)	prove\:\sin(x)\cot(x)+\cos(x)\tan^{2}(x)=\sec(x)
prove (cot^2(x)+1)(1-cos^2(x))=1	prove\:(\cot^{2}(x)+1)(1-\cos^{2}(x))=1
prove sin(x)csc(x)-sin^2(x)=cos^2(x)	prove\:\sin(x)\csc(x)-\sin^{2}(x)=\cos^{2}(x)
prove tan(2pi+θ)=tan(θ)	prove\:\tan(2π+θ)=\tan(θ)
prove (tan(x)-sin(-x))/(1+cos(x))=tan(x)	prove\:\frac{\tan(x)-\sin(-x)}{1+\cos(x)}=\tan(x)
prove cos(a)+sin(a)tan(a)=sec(a)	prove\:\cos(a)+\sin(a)\tan(a)=\sec(a)
prove (csc(θ)+cot(θ))(1-cos(θ))=sin(θ)	prove\:(\csc(θ)+\cot(θ))(1-\cos(θ))=\sin(θ)
prove (1-cos(x))/(sin(x))=csc(x)-cot(x)	prove\:\frac{1-\cos(x)}{\sin(x)}=\csc(x)-\cot(x)
prove cos^2(β)-sin^2(β)=1-2sin^2(β)	prove\:\cos^{2}(β)-\sin^{2}(β)=1-2\sin^{2}(β)
prove csc(2x)= 1/2 sec(x)csc(x)	prove\:\csc(2x)=\frac{1}{2}\sec(x)\csc(x)
prove 1/(1-tan^2(x))+1/(1-cot^2(x))=1	prove\:\frac{1}{1-\tan^{2}(x)}+\frac{1}{1-\cot^{2}(x)}=1
prove tan(2a)=(2tan(a))/(1-tan^2(a))	prove\:\tan(2a)=\frac{2\tan(a)}{1-\tan^{2}(a)}
prove csc(x)-cot(x)= 1/(csc(x)+cot(x))	prove\:\csc(x)-\cot(x)=\frac{1}{\csc(x)+\cot(x)}
prove cos(pi-θ)+sin(pi/2+θ)=0	prove\:\cos(π-θ)+\sin(\frac{π}{2}+θ)=0
prove sin(x)-csc(x)=-cos(x)cot(x)	prove\:\sin(x)-\csc(x)=-\cos(x)\cot(x)
prove tan(y)+cot(y)=sec(y)csc(y)	prove\:\tan(y)+\cot(y)=\sec(y)\csc(y)
prove 1/(sin(x))=csc(x)	prove\:\frac{1}{\sin(x)}=\csc(x)
prove (csc(θ))/(sec(θ))=cot(θ)	prove\:\frac{\csc(θ)}{\sec(θ)}=\cot(θ)
prove sec(x)csc(x)=tan(x)+cot(x)	prove\:\sec(x)\csc(x)=\tan(x)+\cot(x)
prove sin(x)*sec(x)=tan(x)	prove\:\sin(x)\cdot\:\sec(x)=\tan(x)
prove cot(B)sin(B)=cos(B)	prove\:\cot(B)\sin(B)=\cos(B)
prove (cos(x))/(sec(x)-tan(x))=1+sin(x)	prove\:\frac{\cos(x)}{\sec(x)-\tan(x)}=1+\sin(x)
prove (csc^2(x))/(sec^2(x))=csc^2(x)-1	prove\:\frac{\csc^{2}(x)}{\sec^{2}(x)}=\csc^{2}(x)-1
prove 2csc(2x)=csc^2(x)tan(x)	prove\:2\csc(2x)=\csc^{2}(x)\tan(x)
prove (sec^2(t))/(tan(t))=cot(t)+tan(t)	prove\:\frac{\sec^{2}(t)}{\tan(t)}=\cot(t)+\tan(t)
prove csc(pi/2-x)=sec(x)	prove\:\csc(\frac{π}{2}-x)=\sec(x)
prove cos(x)cot(x)=(1-sin^2(x))/(sin(x))	prove\:\cos(x)\cot(x)=\frac{1-\sin^{2}(x)}{\sin(x)}
prove 2sin^2(x)+cos(2x)=1	prove\:2\sin^{2}(x)+\cos(2x)=1
prove 2cot(4x)=cot(2x)-tan(2x)	prove\:2\cot(4x)=\cot(2x)-\tan(2x)
prove sin(2x)-cot(x)=-cot(x)cos(2x)	prove\:\sin(2x)-\cot(x)=-\cot(x)\cos(2x)
prove (1+sec(x))/(sin(x)+tan(x))=csc(x)	prove\:\frac{1+\sec(x)}{\sin(x)+\tan(x)}=\csc(x)
prove (tan^2(x))/(1+tan^2(x))=sin^2(x)	prove\:\frac{\tan^{2}(x)}{1+\tan^{2}(x)}=\sin^{2}(x)
prove (sin(x))/(cos(x))=tan(x)	prove\:\frac{\sin(x)}{\cos(x)}=\tan(x)
prove tan(pi/2-x)sec(x)=csc(x)	prove\:\tan(\frac{π}{2}-x)\sec(x)=\csc(x)
prove 1/(cos(x))-cos(x)=sin(x)tan(x)	prove\:\frac{1}{\cos(x)}-\cos(x)=\sin(x)\tan(x)
prove sec(u)cos(u)-sin^2(u)=cos^2(u)	prove\:\sec(u)\cos(u)-\sin^{2}(u)=\cos^{2}(u)
prove 1+cot^2(x/2)= 2/(sin(x)tan(x/2))	prove\:1+\cot^{2}(\frac{x}{2})=\frac{2}{\sin(x)\tan(\frac{x}{2})}
prove (tan^2(x))/(sec(x))+cos(x)=sec(x)	prove\:\frac{\tan^{2}(x)}{\sec(x)}+\cos(x)=\sec(x)
prove tan^2(x)cot^2(x)=1	prove\:\tan^{2}(x)\cot^{2}(x)=1
prove cos(pi+θ)=-cos(θ)	prove\:\cos(π+θ)=-\cos(θ)
prove cos((3pi)/2-θ)=-sin(θ)	prove\:\cos(\frac{3π}{2}-θ)=-\sin(θ)
prove 1+tan^2(x)= 1/(cos^2(x))	prove\:1+\tan^{2}(x)=\frac{1}{\cos^{2}(x)}
prove cos(B)csc(B)tan(B)=1	prove\:\cos(B)\csc(B)\tan(B)=1
prove (tan(θ)+cot(θ))/(tan(θ))=csc^2(θ)	prove\:\frac{\tan(θ)+\cot(θ)}{\tan(θ)}=\csc^{2}(θ)
prove sin(60)=2sin(30)cos(30)	prove\:\sin(60^{\circ\:})=2\sin(30^{\circ\:})\cos(30^{\circ\:})
prove (cos(θ))/(1+sin(θ))+tan(θ)=sec(θ)	prove\:\frac{\cos(θ)}{1+\sin(θ)}+\tan(θ)=\sec(θ)
prove cos(x)*tan(x)=sin(x)	prove\:\cos(x)\cdot\:\tan(x)=\sin(x)
prove (sin(2θ))/(sin(θ))-(cos(2θ))/(cos(θ))=sec(θ)	prove\:\frac{\sin(2θ)}{\sin(θ)}-\frac{\cos(2θ)}{\cos(θ)}=\sec(θ)
prove (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x)	prove\:\frac{\sec(x)}{\csc(x)}+\frac{\sin(x)}{\cos(x)}=2\tan(x)
prove (sin(θ)+tan(θ))/(1+sec(θ))=sin(θ)	prove\:\frac{\sin(θ)+\tan(θ)}{1+\sec(θ)}=\sin(θ)
prove sec(2θ)=(sec^2(θ))/(1-tan^2(θ))	prove\:\sec(2θ)=\frac{\sec^{2}(θ)}{1-\tan^{2}(θ)}
prove cos(2a)=cos^2(a)-sin^2(a)	prove\:\cos(2a)=\cos^{2}(a)-\sin^{2}(a)
prove tan(θ)+cot(θ)=sec(θ)*csc(θ)	prove\:\tan(θ)+\cot(θ)=\sec(θ)\cdot\:\csc(θ)
prove (cos(x))/(csc(x)+1)+(cos(x))/(csc(x)-1)=2tan(x)	prove\:\frac{\cos(x)}{\csc(x)+1}+\frac{\cos(x)}{\csc(x)-1}=2\tan(x)
prove (cos(x)csc(x))/(cot^2(x))=tan(x)	prove\:\frac{\cos(x)\csc(x)}{\cot^{2}(x)}=\tan(x)
prove sin^2(θ)-cos^2(θ)=2sin^2(θ)-1	prove\:\sin^{2}(θ)-\cos^{2}(θ)=2\sin^{2}(θ)-1
prove cot(pi/2-u)=tan(u)	prove\:\cot(\frac{π}{2}-u)=\tan(u)
prove (sec^4(x)-1)/(tan^2(x))=tan^2(x)+2	prove\:\frac{\sec^{4}(x)-1}{\tan^{2}(x)}=\tan^{2}(x)+2
prove 1/(sec^2(x))=cos^2(x)	prove\:\frac{1}{\sec^{2}(x)}=\cos^{2}(x)
prove tan(x)+cot(x)= 1/(sin(x)cos(x))	prove\:\tan(x)+\cot(x)=\frac{1}{\sin(x)\cos(x)}
prove (cot^2(x))/(csc(x)-1)=csc(x)+1	prove\:\frac{\cot^{2}(x)}{\csc(x)-1}=\csc(x)+1
prove tan^2(x)-sec^2(x)=-1	prove\:\tan^{2}(x)-\sec^{2}(x)=-1
prove tan^2(θ)-sin^2(θ)=sin^4(θ)sec^2(θ)	prove\:\tan^{2}(θ)-\sin^{2}(θ)=\sin^{4}(θ)\sec^{2}(θ)
prove cos(-θ)=cos(θ)	prove\:\cos(-θ)=\cos(θ)
prove 1+sin(2θ)=(sin(θ)+cos(θ))^2	prove\:1+\sin(2θ)=(\sin(θ)+\cos(θ))^{2}
prove cos((5pi)/4-x)=-(sqrt(2))/2 (cos(x)+sin(x))	prove\:\cos(\frac{5π}{4}-x)=-\frac{\sqrt{2}}{2}(\cos(x)+\sin(x))
prove (sin^2(x))/(cos(x))=sec(x)-cos(x)	prove\:\frac{\sin^{2}(x)}{\cos(x)}=\sec(x)-\cos(x)
prove 5cos^2(θ)+3sin^2(θ)=3+2cos^2(θ)	prove\:5\cos^{2}(θ)+3\sin^{2}(θ)=3+2\cos^{2}(θ)
prove cos(x+pi/3)+cos(x-pi/3)=cos(x)	prove\:\cos(x+\frac{π}{3})+\cos(x-\frac{π}{3})=\cos(x)
prove 3cos(x+y)+3cos(x-y)=6cos(x)cos(y)	prove\:3\cos(x+y)+3\cos(x-y)=6\cos(x)\cos(y)
prove cos(x)=2cos^2(x/2)-1	prove\:\cos(x)=2\cos^{2}(\frac{x}{2})-1
prove tan(x+pi/4)=(tan(x)+1)/(1-tan(x))	prove\:\tan(x+\frac{π}{4})=\frac{\tan(x)+1}{1-\tan(x)}
prove csc(θ)cos(θ)tan(θ)=1	prove\:\csc(θ)\cos(θ)\tan(θ)=1
prove sin(θ)cos(φ)=(sin(θ+φ)+sin(θ-φ))/2	prove\:\sin(θ)\cos(φ)=\frac{\sin(θ+φ)+\sin(θ-φ)}{2}
prove (cos(x)-sin(x))^2=1-2cos(x)sin(x)	prove\:(\cos(x)-\sin(x))^{2}=1-2\cos(x)\sin(x)
prove (cos(x))/(sin(x))=cos^2(x)	prove\:\frac{\cos(x)}{\sin(x)}=\cos^{2}(x)
prove sin(α+β)+sin(α-β)=2sin(α)cos(β)	prove\:\sin(α+β)+\sin(α-β)=2\sin(α)\cos(β)
prove tan(x)=(sin(2x))/(1+cos(2x))	prove\:\tan(x)=\frac{\sin(2x)}{1+\cos(2x)}
prove 2sin(x)cos(x)=sin(2x)	prove\:2\sin(x)\cos(x)=\sin(2x)
prove cos(x)tan^2(x)+cos(x)=sec(x)	prove\:\cos(x)\tan^{2}(x)+\cos(x)=\sec(x)
prove 1-sin(x)=cos(x)	prove\:1-\sin(x)=\cos(x)
prove (cos(2x)+sin(2x))^2=1+sin(4x)	prove\:(\cos(2x)+\sin(2x))^{2}=1+\sin(4x)
prove cos^2(x)csc(x)-csc(x)=-sin(x)	prove\:\cos^{2}(x)\csc(x)-\csc(x)=-\sin(x)
prove (cos(θ))/(sec(θ)-tan(θ))=1+sin(θ)	prove\:\frac{\cos(θ)}{\sec(θ)-\tan(θ)}=1+\sin(θ)

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