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

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

0.39mins
Q1

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

0.43mins
Q2

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

0.25mins
Q3

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

0.44mins
Q4

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

0.26mins
Q5

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

0.46mins
Q6

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

0.23mins
Q7

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

0.40mins
Q8

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

1.27mins
Q9a

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

0.50mins
Q10a

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

0.48mins
Q10b

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

2.26mins
Q11

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

0.40mins
Q12

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

1.30mins
Q13

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

0.57mins
Q15

Prove that

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

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.

0.14mins
Q19ab

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

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

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.

3.13mins
Q20b

Prove that

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

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