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Popular Trigonometry Problems

prove sin^{0.5}(x)cos(x)-sin^{2.5}(x)cos(x)=cos^3(x)sin(x)	prove\:\sin^{0.5}(x)\cos(x)-\sin^{2.5}(x)\cos(x)=\cos^{3}(x)\sin(x)
prove 1-2cos^2(x)=2sin^2(x)	prove\:1-2\cos^{2}(x)=2\sin^{2}(x)
prove sin((2pi)/3)=(sqrt(3))/2	prove\:\sin(\frac{2π}{3})=\frac{\sqrt{3}}{2}
prove 1/2 2sin(x)cos(x)= 1/2 sin(2x)	prove\:\frac{1}{2}2\sin(x)\cos(x)=\frac{1}{2}\sin(2x)
prove 1-tan^2(x)=sec^2(x)	prove\:1-\tan^{2}(x)=\sec^{2}(x)
prove (cot(x)+1)/(csc(x))=cos(x)+sin(x)	prove\:\frac{\cot(x)+1}{\csc(x)}=\cos(x)+\sin(x)
prove (sec(θ))/(sin(θ))=cot(θ)+tan(θ)	prove\:\frac{\sec(θ)}{\sin(θ)}=\cot(θ)+\tan(θ)
prove (sin(2a)+sin(4a))/(cos(2a)-cos(4a))=cot(a)	prove\:\frac{\sin(2a)+\sin(4a)}{\cos(2a)-\cos(4a)}=\cot(a)
prove (tan(a))/(sec(a))=tan(a)cos(a)	prove\:\frac{\tan(a)}{\sec(a)}=\tan(a)\cos(a)
prove (cos(θ))/(1+sin(θ))=sec(θ)tan(θ)	prove\:\frac{\cos(θ)}{1+\sin(θ)}=\sec(θ)\tan(θ)
prove (tan(X)-sin(X))/(sin^3(X))=(sec(X))/(1+cos(X))	prove\:\frac{\tan(X)-\sin(X)}{\sin^{3}(X)}=\frac{\sec(X)}{1+\cos(X)}
prove (tan(x)-1)/(1-cot(x))=tan(x)	prove\:\frac{\tan(x)-1}{1-\cot(x)}=\tan(x)
prove sin(x)+cos(x)=(cot(x)+1)/(csc(x))	prove\:\sin(x)+\cos(x)=\frac{\cot(x)+1}{\csc(x)}
prove sin^2(t)=2sin(t)cos(t)	prove\:\sin^{2}(t)=2\sin(t)\cos(t)
prove sin((2pi)/3)=sin((-11pi)/3)	prove\:\sin(\frac{2π}{3})=\sin(\frac{-11π}{3})
prove (sin(2x))/(tan(x))=2cos^2(x)	prove\:\frac{\sin(2x)}{\tan(x)}=2\cos^{2}(x)
prove cos^2(5θ)-sin^2(5θ)=cos(10θ)	prove\:\cos^{2}(5θ)-\sin^{2}(5θ)=\cos(10θ)
prove 1-cos(θ)=(sin^2(θ))/(1+cos(θ))	prove\:1-\cos(θ)=\frac{\sin^{2}(θ)}{1+\cos(θ)}
prove (sin^4(θ)-cos^4(θ))/(sin^2(θ)cos^2(θ))=sec^2(θ)-csc^2(θ)	prove\:\frac{\sin^{4}(θ)-\cos^{4}(θ)}{\sin^{2}(θ)\cos^{2}(θ)}=\sec^{2}(θ)-\csc^{2}(θ)
prove (sin(2x))/(sin(x))= 2/(sec(x))	prove\:\frac{\sin(2x)}{\sin(x)}=\frac{2}{\sec(x)}
prove 1-tanh^2(z)=sech^2(z)	prove\:1-\tanh^{2}(z)=\sech^{2}(z)
prove csc(s)=cot(s)sec(s)	prove\:\csc(s)=\cot(s)\sec(s)
prove csc^4(x)-cot^4(x)=2sec^2(x)-1	prove\:\csc^{4}(x)-\cot^{4}(x)=2\sec^{2}(x)-1
prove 2/(1+cos(2x))=sec^2(x)	prove\:\frac{2}{1+\cos(2x)}=\sec^{2}(x)
prove cos^2(x)-sin^2(x)=2cos^2(-1)	prove\:\cos^{2}(x)-\sin^{2}(x)=2\cos^{2}(-1)
prove tan^2(x)=(sec(x)-1)(sec(x)+1)	prove\:\tan^{2}(x)=(\sec(x)-1)(\sec(x)+1)
prove cos(2x)=-2sin(x^2)+1	prove\:\cos(2x)=-2\sin(x^{2})+1
prove tan(x)(cos(x)+cot(x))=sin(x)+1	prove\:\tan(x)(\cos(x)+\cot(x))=\sin(x)+1
prove tan^2(x)-sin^2(x)tan^2(x)=sin^2(x)	prove\:\tan^{2}(x)-\sin^{2}(x)\tan^{2}(x)=\sin^{2}(x)
prove (1-cos(x))(1+cos(x))(1+cot^2(x))=1	prove\:(1-\cos(x))(1+\cos(x))(1+\cot^{2}(x))=1
prove 1+tan^2(60)=sec^2(60)	prove\:1+\tan^{2}(60^{\circ\:})=\sec^{2}(60^{\circ\:})
prove 1+cos(θ)=sin(θ)	prove\:1+\cos(θ)=\sin(θ)
prove sin(x)=sin(x+2pi)	prove\:\sin(x)=\sin(x+2π)
prove 1+cos(t)=(sin^2(t))/(1-cos(t))	prove\:1+\cos(t)=\frac{\sin^{2}(t)}{1-\cos(t)}
prove cos(3x)=cos(x)(1-4sin^2(x))	prove\:\cos(3x)=\cos(x)(1-4\sin^{2}(x))
prove csc(x)cos(x)=cos^2(x)	prove\:\csc(x)\cos(x)=\cos^{2}(x)
prove tan(A)+cot(A)= 2/(sin(2A))	prove\:\tan(A)+\cot(A)=\frac{2}{\sin(2A)}
prove (1-cos^2(x))/(1+sin(x))=sin(x)	prove\:\frac{1-\cos^{2}(x)}{1+\sin(x)}=\sin(x)
prove cos(θ)tan(θ)csc(θ)=1	prove\:\cos(θ)\cdot\:\tan(θ)\cdot\:\csc(θ)=1
prove csc(x)=(cos(x)cot(x))/(1-sin(x))-1	prove\:\csc(x)=\frac{\cos(x)\cot(x)}{1-\sin(x)}-1
prove 1/2 cot(x/2)-1/2 tan(x/2)=cot(x)	prove\:\frac{1}{2}\cot(\frac{x}{2})-\frac{1}{2}\tan(\frac{x}{2})=\cot(x)
prove 1-sin^2(x)=(1-sin(x))(1+sin(x))	prove\:1-\sin^{2}(x)=(1-\sin(x))(1+\sin(x))
prove (-cot(x)+1)/(tan(x)-1)=cot(x)	prove\:\frac{-\cot(x)+1}{\tan(x)-1}=\cot(x)
prove 1/2 (-cos(1)+1)=(1-cos(1))/2	prove\:\frac{1}{2}(-\cos(1)+1)=\frac{1-\cos(1)}{2}
prove 4cos^2(a)+4sin^2(a)=4	prove\:4\cos^{2}(a)+4\sin^{2}(a)=4
prove (sin(a))/(tan(a))=cos(a)	prove\:\frac{\sin(a)}{\tan(a)}=\cos(a)
prove 4sin^2(θ)+8cos(θ)=3	prove\:4\sin^{2}(θ)+8\cos(θ)=3
prove sin^2(3x)+cos^2(3x)=1	prove\:\sin^{2}(3x)+\cos^{2}(3x)=1
prove csc(a)-sin(a)=cos(a)cot(a)	prove\:\csc(a)-\sin(a)=\cos(a)\cot(a)
prove cos^2(α)+cos^2(α)tan^2(α)=1	prove\:\cos^{2}(α)+\cos^{2}(α)\tan^{2}(α)=1
prove cos(pi/7)=sin(pi/2-pi/7)	prove\:\cos(\frac{π}{7})=\sin(\frac{π}{2}-\frac{π}{7})
prove sin(pi/4+x)=sqrt(2)(cos(x)+sin(x))	prove\:\sin(\frac{π}{4}+x)=\sqrt{2}(\cos(x)+\sin(x))
prove sec(x)csc(x)=cot(x)+tan(x)	prove\:\sec(x)\csc(x)=\cot(x)+\tan(x)
prove arctan(x)=(arcsin(x))/(arccos(x))	prove\:\arctan(x)=\frac{\arcsin(x)}{\arccos(x)}
prove (cos(θ/2)-sin(θ/2))^2=1-sin(θ)	prove\:(\cos(\frac{θ}{2})-\sin(\frac{θ}{2}))^{2}=1-\sin(θ)
prove sqrt(1+sin(4a)+sin(2a))=-2cos(a)	prove\:\sqrt{1+\sin(4a)+\sin(2a)}=-2\cos(a)
prove (csc^2(θ))(tan^2(θ))-1=tan^2(θ)	prove\:(\csc^{2}(θ))(\tan^{2}(θ))-1=\tan^{2}(θ)
prove cot(x)-cot(2x)=csc(2x)	prove\:\cot(x)-\cot(2x)=\csc(2x)
prove (csc(θ)+1)/(1+sin(θ))=csc(θ)	prove\:\frac{\csc(θ)+1}{1+\sin(θ)}=\csc(θ)
prove (tan(a)sin(a))/(1-cos(a))=sec(a)+1	prove\:\frac{\tan(a)\sin(a)}{1-\cos(a)}=\sec(a)+1
prove sec(A)csc(A)=csc(A)tan(A)	prove\:\sec(A)\csc(A)=\csc(A)\tan(A)
prove (sin^2(θ))/(cos^2(θ))=sec^2(θ)-1	prove\:\frac{\sin^{2}(θ)}{\cos^{2}(θ)}=\sec^{2}(θ)-1
prove sin(x)*tan(x)+cos(-x)=sec(x)	prove\:\sin(x)\cdot\:\tan(x)+\cos(-x)=\sec(x)
prove cos(2α)=cos^2(α)-sin^2(α)	prove\:\cos(2α)=\cos^{2}(α)-\sin^{2}(α)
prove (tan(x))/(1+tan^2(x))=cos(x)sin(x)	prove\:\frac{\tan(x)}{1+\tan^{2}(x)}=\cos(x)\sin(x)
prove 2csc(2x)= 1/(sin(x)cos(x))	prove\:2\csc(2x)=\frac{1}{\sin(x)\cos(x)}
prove sin(u)(csc(u)-sin(u))=cos^2(u)	prove\:\sin(u)(\csc(u)-\sin(u))=\cos^{2}(u)
prove cos^2(θ)-sin^2(θ)=1	prove\:\cos^{2}(θ)-\sin^{2}(θ)=1
prove (cos(θ))/(1-sin^2(θ))= 1/(cos(θ))	prove\:\frac{\cos(θ)}{1-\sin^{2}(θ)}=\frac{1}{\cos(θ)}
prove (1-tan^4(θ))/(sec^2(θ))=1-tan^2(θ)	prove\:\frac{1-\tan^{4}(θ)}{\sec^{2}(θ)}=1-\tan^{2}(θ)
prove (1+csc(θ))(1-sin(θ))=csc(θ)-sin(θ)	prove\:(1+\csc(θ))(1-\sin(θ))=\csc(θ)-\sin(θ)
prove (2cot(x))/(1+cot^2(x))=cos(2x)	prove\:\frac{2\cot(x)}{1+\cot^{2}(x)}=\cos(2x)
prove sin^2(x)cos^2(x)= 1/8 (1-sin(4x))	prove\:\sin^{2}(x)\cos^{2}(x)=\frac{1}{8}(1-\sin(4x))
prove (1-tan(x))/(1-cot(x))=-tan(x)	prove\:\frac{1-\tan(x)}{1-\cot(x)}=-\tan(x)
prove (sin(A)+cos(A))^2=1+2sin(A)cos(A)	prove\:(\sin(A)+\cos(A))^{2}=1+2\sin(A)\cos(A)
prove (cos(b))/(sec(b))+(sin(b))/(csc(b))=csc^2(b)-cot^2(b)	prove\:\frac{\cos(b)}{\sec(b)}+\frac{\sin(b)}{\csc(b)}=\csc^{2}(b)-\cot^{2}(b)
prove 2cos^2(θ/2)-cos(θ)-1=0	prove\:2\cos^{2}(\frac{θ}{2})-\cos(θ)-1=0
prove 1/(csc(θ)-cot(θ))=csc(θ)+cot(θ)	prove\:\frac{1}{\csc(θ)-\cot(θ)}=\csc(θ)+\cot(θ)
prove 1+cot^2(-x)=csc^2(x)	prove\:1+\cot^{2}(-x)=\csc^{2}(x)
prove cos(θ+pi/4)=(sqrt(2))/2 (cos(θ)-sin(θ))	prove\:\cos(θ+\frac{π}{4})=\frac{\sqrt{2}}{2}(\cos(θ)-\sin(θ))
prove 5cos^2(x)+7sin^2(x)-5=2sin^2(x)	prove\:5\cos^{2}(x)+7\sin^{2}(x)-5=2\sin^{2}(x)
prove sin(θ)-sin(θ-2pi)=0	prove\:\sin(θ)-\sin(θ-2π)=0
prove-sin(2x)=-2sin(x)cos(x)	prove\:-\sin(2x)=-2\sin(x)\cos(x)
prove csc(2x)=(sec(x)csc(x))/2	prove\:\csc(2x)=\frac{\sec(x)\csc(x)}{2}
prove sec(2x)=((sec^2(x)))/(2-sec^2(x))	prove\:\sec(2x)=\frac{(\sec^{2}(x))}{2-\sec^{2}(x)}
prove sin(x)=0	prove\:\sin(x)=0
prove sin(x)=1	prove\:\sin(x)=1
prove sin(θ)tan(θ)=(1-cos^2(θ))/(cos(θ))	prove\:\sin(θ)\tan(θ)=\frac{1-\cos^{2}(θ)}{\cos(θ)}
prove sin(x+30)+sqrt(3)cos(x+30)=2cos(x)	prove\:\sin(x+30^{\circ\:})+\sqrt{3}\cos(x+30^{\circ\:})=2\cos(x)
prove (sec(a)+tan(a))(1-sin(a))=cos(a)	prove\:(\sec(a)+\tan(a))(1-\sin(a))=\cos(a)
prove tan^2(a)+1=sec^2(a)	prove\:\tan^{2}(a)+1=\sec^{2}(a)
prove (8csc(-x))/(sec(-x))=-8cot(x)	prove\:\frac{8\csc(-x)}{\sec(-x)}=-8\cot(x)
prove cos(α-β)=cos(α)cos(β)+sin(α)sin(β)	prove\:\cos(α-β)=\cos(α)\cos(β)+\sin(α)\sin(β)
prove (4sin^2(t))/(tan^2(t))=4cos^2(t)	prove\:\frac{4\sin^{2}(t)}{\tan^{2}(t)}=4\cos^{2}(t)
prove (csc^2(t))/(csc^2(t)-2)=sec(2t)	prove\:\frac{\csc^{2}(t)}{\csc^{2}(t)-2}=\sec(2t)
prove cos^2(x)-sin^2(x)=1-sin^2(x)	prove\:\cos^{2}(x)-\sin^{2}(x)=1-\sin^{2}(x)
prove sin(x)*tan(x)+cos(x)=sec(x)	prove\:\sin(x)\cdot\:\tan(x)+\cos(x)=\sec(x)
prove cot(α-β)=(cot(α)cot(β)+1)/(cot(β)-cot(α))	prove\:\cot(α-β)=\frac{\cot(α)\cot(β)+1}{\cot(β)-\cot(α)}
prove tan(2x)=tan(x+x)	prove\:\tan(2x)=\tan(x+x)
prove sin^3(x)= 3/4 sin(x)-1/4 sin(3x)	prove\:\sin^{3}(x)=\frac{3}{4}\sin(x)-\frac{1}{4}\sin(3x)

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