4.5 Prove Trigonometric Identities
Chapter
Chapter 4
Section
4.5
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Solutions 22 Videos

Prove that \cos2x=2\cos^2x-1.

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0.39mins
Q1

Prove that \cos2x=1-2\sin^2x.

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0.43mins
Q2

Prove that \sin(x+\pi)=-\sin x.

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0.25mins
Q3

Prove that \displaystyle{\sin\left(\frac{3\pi}{2}-x\right)=-\cos x}.

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0.44mins
Q4

Prove that \cos(\pi-x)=-\cos x.

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0.26mins
Q5

Prove that \displaystyle{\cos\left(\frac{3\pi}{2}+x\right)=\sin x}.

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0.46mins
Q6

Prove that \cos x=\sin x\cot x.

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0.23mins
Q7

Prove that 1+\sin x=\sin x(1+\csc x).

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0.40mins
Q8

Prove that 1-2\cos^2x=\sin x\cos x(\tan x-\cot x).

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1.27mins
Q9a

Prove that \csc^2x=1+\cot^2x.

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0.50mins
Q10a

Prove that \sec^2x=1+\tan^2x.

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0.48mins
Q10b

Prove that \displaystyle{\frac{1-\sin^2x}{\cos x}=\frac{\sin2x}{2\sin x}}.

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2.26mins
Q11

Prove that \displaystyle{\frac{\csc^2x-1}{\csc^2x}=1-\sin^2x}

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0.40mins
Q12

Prove that \displaystyle{\frac{\csc x}{\cos x}=\tan x+\cot x}.

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1.30mins
Q13

Prove that 2\sin x\sin y=\cos(x-y)-\cos(x+y).

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0.57mins
Q15

Prove that

\sin2x+\sin2y=2\sin(x+y)\cos(x-y).

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1.22mins
Q16

Use an appropriate compound angle formula to determine to determine an expression for \sin3x in terms of \sin x and \cos x

(a) Write 3x as the sum of two terms involving x.

(b) Substitute the expression from part (a) into sin 3x.

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0.14mins
Q19ab

Use an appropriate compound angle formula to expand the expression for \sin(x + 2x).

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0.12mins
Q19c

Use an appropriate compound angle formula to determine to determine an expression for \sin3x in terms of \sin x and \cos x

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0.58mins
Q19d

(a) Use graphing technology to determine whether it is reasonable to conjecture that \sin^6x+\cos^6x=1-3\sin^2x\cos^2x is an identity.

(b) If it appears to be an identity, prove the identity. If not, determine a counterexample.

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3.13mins
Q20b

Prove that

\cos^4x-\sin^4x=\cos2x.

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0.48mins
Q21

Prove

\cos(\sin^{-1}x)=\sqrt{1-x^2}

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0.46mins
Q23a