Proving Trigonometric Identities

Question 1 of 6

Identify if the equation is correct

\frac{cos θ}{1 - sin θ} = sec θ + tan θ

$(cos theta)/(1-sin theta)=sec theta+tan theta$

1. $Correct$ $\text(Correct)$
2. $Incorrect$ $\text(Incorrect)$

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Trigonometric Identities

sec θ = \frac{1}{cos θ}

$\sec\theta=\frac{1}{\text{cos}\theta}$

csc θ = \frac{1}{sin θ}

$\csc\theta=\frac{1}{\text{sin}\theta}$

cot θ = \frac{1}{tan θ}

$\cot\theta=\frac{1}{\text{tan}\theta}$

tan θ = \frac{sin θ}{cos θ}

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

cot θ = \frac{cos θ}{sin θ}

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

	$\frac{cos θ}{1 - sin θ}$ $(\text(cos) theta)/(1-\text(sin) theta)$

$=$ $=$	$\frac{\text{cos}\;\theta\cdot(\color{#CC0000}{1+\text{sin}\;\theta})}{1-\text{sin}\;\theta\cdot(\color{#CC0000}{1+\text{sin}\;\theta})}$	Multiply the numerator and denominator by $1 + sin θ$ $1+\text(sin) theta$

$=$ $=$	$\frac{cos θ (1 + sin θ)}{1 - {sin}^{2} θ}$ $(\text(cos) theta(1+\text(sin) theta))/(1-\text(sin)^2 theta)$	Difference of two squares

Recall that

{sin}^{2} θ + {cos}^{2} θ = 1

$\text(sin)^2theta+\text(cos)^2theta=1$ . Derive this formula to further simplify the expression

${sin}^{2} θ + {cos}^{2} θ$ $\text(sin)^2theta+\text(cos)^2theta$	$=$ $=$	$1$ $1$
${sin}^{2} θ + {cos}^{2} θ$ $\text(sin)^2theta+\text(cos)^2theta$ $- {sin}^{2} θ$ $-\text(sin)^2theta$	$=$ $=$	$1$ $1$ $- {sin}^{2} θ$ $-\text(sin)^2theta$	Subtract ${sin}^{2} θ$ $\text(sin)^2theta$ from both sides
${cos}^{2} θ$ $\text(cos)^2theta$	$=$ $=$	$1 - {sin}^{2} θ$ $1-\text(sin)^2theta$

	$\frac{cos θ (1 + sin θ)}{1 - {sin}^{2} θ}$ $(\text(cos) theta(1+\text(sin) theta))/(1-\text(sin)^2 theta)$

$=$ $=$	$\frac{cos θ (1 + sin θ)}{{cos}^{2} θ}$ $(\text(cos) theta(1+\text(sin) theta))/(\text(cos)^2theta)$	$1 - {sin}^{2} θ = {cos}^{2} θ$ $1-\text(sin)^2theta=\text(cos)^2theta$

$=$ $=$	$\frac{1 + sin θ}{cos θ}$ $(1+\text(sin) theta)/(\text(cos) theta)$	$\frac{cos θ}{{cos}^{2} θ} = \frac{1}{cos θ}$ $(\text(cos) theta)/(\text(cos)^2theta)=1/(\text(cos) theta)$

$=$ $=$	$\frac{1}{cos θ} + \frac{sin θ}{cos θ}$ $1/(\text(cos) theta)+(\text(sin) theta)/(\text(cos)theta)$	Separate the values of the numerator

$=$ $=$	$sec θ + \frac{sin θ}{cos θ}$ $\text(sec)theta+(\text(sin) theta)/(\text(cos)theta)$	$\frac{1}{cos θ} = sec θ$ $1/(\text(cos) theta)=\text(sec)theta$

$=$ $=$	$sec θ + tan θ$ $\text(sec)theta+\text(tan)theta$	$\frac{sin θ}{cos θ} = tan θ$ $(\text(sin) theta)/(\text(cos)theta)=\text(tan)theta$

This means that

\frac{cos θ}{1 - sin θ} = sec θ + tan θ

$(\text(cos) theta)/(1-\text(sin) theta)=\text(sec) theta+\text(tan) theta$

Correct

$\text(Correct)$

Question 2 of 6

Identify if the equation is correct

\frac{sin A}{1 + cos A} + \frac{1 + cos A}{sin A} = 2 csc A

$(sin A)/(1+cos A)+(1+cos A)/(sin A)=2 csc A$

1. $Incorrect$ $\text(Incorrect)$
2. $Correct$ $\text(Correct)$

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Trigonometric Identities

sec θ = \frac{1}{cos θ}

$\sec\theta=\frac{1}{\text{cos}\theta}$

csc θ = \frac{1}{sin θ}

$\csc\theta=\frac{1}{\text{sin}\theta}$

cot θ = \frac{1}{tan θ}

$\cot\theta=\frac{1}{\text{tan}\theta}$

tan θ = \frac{sin θ}{cos θ}

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

cot θ = \frac{cos θ}{sin θ}

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

	$\frac{sin A}{1 + cos A} + \frac{1 + cos A}{sin A}$ $(\text(sin) A)/(1+\text(cos) A)+(1+\text(cos) A)/(\text(sin) A)$

$=$ $=$	$\frac{{sin}^{2} A + {(1 + cos A)}^{2}}{(1 + cos A) sin A}$ $(\text(sin)^2A+(1+\text(cos) A)^2)/((1+\text(cos) A)\text(sin) A)$	Apply the rule of adding fractions

$=$ $=$	$\frac{{sin}^{2} A + (1 + cos A) (1 + cos A)}{(1 + cos A) sin A}$ $(\text(sin)^2A+(1+\text(cos) A)(1+\text(cos) A))/((1+\text(cos) A)\text(sin) A)$	Expand

$=$ $=$	$\frac{{sin}^{2} A + 1 + 2 cos A + {cos}^{2} A}{(1 + cos A) sin A}$ $(\text(sin)^2A+1+2\text(cos) A+\text(cos)^2A)/((1+\text(cos) A)\text(sin) A)$

Recall that

{sin}^{2} θ + {cos}^{2} θ = 1

$\text(sin)^2theta+\text(cos)^2theta=1$ . Derive this formula to further simplify the expression

	$\frac{{sin}^{2} A + 1 + 2 cos A + {cos}^{2} A}{(1 + cos A) sin A}$ $(\text(sin)^2A+1+2\text(cos) A+\text(cos)^2A)/((1+\text(cos) A)\text(sin) A)$

$=$ $=$	$\frac{1 + 1 + 2 cos A}{(1 + cos A) sin A}$ $(1+1+2\text(cos) A)/((1+\text(cos) A)\text(sin) A)$	${sin}^{2} A + {cos}^{2} A = 1$ $\text(sin)^2A+\text(cos)^2A=1$

$=$ $=$	$\frac{2 + 2 cos A}{(1 + cos A) sin A}$ $(2+2\text(cos) A)/((1+\text(cos) A)\text(sin) A)$	Combine like terms

$=$ $=$	$\frac{2 (1 + cos A)}{(1 + cos A) sin A}$ $(2(1+\text(cos) A))/((1+\text(cos) A)\text(sin) A)$	Factor out $2$ $2$

$=$ $=$	$\frac{2}{sin A}$ $2/(\text(sin) A)$	$\frac{1 + cos A}{1 + cos A} = 1$ $(1+\text(cos) A)/(1+\text(cos) A)=1$

$=$ $=$	$2 csc A$ $2\text(csc) A$	$\frac{1}{sin A} = csc A$ $1/(\text(sin) A)=\text(csc) A$

This means that

\frac{sin A}{1 + cos A} + \frac{1 + cos A}{sin A} = 2 csc A

$(sin A)/(1+cos A)+(1+cos A)/(sin A)=2 csc A$

Correct

$\text(Correct)$

Question 3 of 6

Identify if the equation is correct

\frac{1 + cot θ}{csc θ} - \frac{sec θ}{tan θ + cot θ} = cos θ

$(1+cot theta)/(csc theta)-(sec theta)/(tan theta+cot theta)=cos theta$

1. $Incorrect$ $\text(Incorrect)$
2. $Correct$ $\text(Correct)$

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Trigonometric Identities

sec θ = \frac{1}{cos θ}

$\sec\theta=\frac{1}{\text{cos}\theta}$

csc θ = \frac{1}{sin θ}

$\csc\theta=\frac{1}{\text{sin}\theta}$

cot θ = \frac{1}{tan θ}

$\cot\theta=\frac{1}{\text{tan}\theta}$

tan θ = \frac{sin θ}{cos θ}

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

cot θ = \frac{cos θ}{sin θ}

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

First term

	$\frac{1 + cot θ}{csc θ}$ $(1+\text(cot) theta)/(\text(csc) theta)$

$=$ $=$	$1/(\text(csc) theta)(1+\text(cot) theta)$	Factorise

$=$	$\text(sin) theta(1+\text(cot) theta)$	$1/(\text(csc) theta)=\text(sin) theta$

$=$	$\text(sin) theta(1+(\text(cos) theta)/(\text(sin) theta))$	$\text(cot) theta=(\text(cos) theta)/(\text(sin) theta)$

$=$	$\text(sin) theta+(\text(sin) theta*\text(cos) theta)/(\text(sin) theta)$	Expand

$=$	$\text(sin) theta+\text(cos) theta$	$(\text(sin) theta)/(\text(sin) theta)=1$

Second term

	$(\text(sec) theta)/(\text(tan) theta+\text(cot) theta)$

$=$	$(1/(\text(cos) theta))/(\text(tan) theta+\text(cot) theta)$	$\text(sec) theta=1/(\text(cos) theta)$

$=$	$(1/(\text(cos) theta))/((\text(sin) theta)/(\text(cos) theta)+\text(cot) theta)$	$\text(tan) theta=(\text(sin) theta)/(\text(cos) theta)$

$=$	$(1/(\text(cos) theta))/((\text(sin) theta)/(\text(cos) theta)+(\text(cos) theta)/(\text(sin) theta)$	$\text(cot) theta=(\text(cos) theta)/(\text(sin) theta)$

$=$	$(1/(\text(cos) theta))/((\text(sin)^2theta+\text(cos)^2theta)/(\text(cos) theta \text(sin) theta)$	Apply rule of adding fractions to the denominator

$=$	$(1/(\text(cos) theta))/(1/(\text(cos) theta \text(sin) theta)$	$\text(sin)^2theta+\text(cos)^2theta=1$

$=$	$1/(\text(cos) theta)*(\text(cos) theta \text(sin) theta)/1$	Apply rule of dividing fractions

$=$	$\text(sin) theta$

Finally, combine the two terms and subtract

		$\text(sin) theta+\text(cos) theta$ $-$ $\text(sin) theta$
	$=$	$\text(cos) theta$

This means that

$(1+\text(cot) theta)/(\text(csc) theta)-(\text(sec) theta)/(\text(tan) theta+\text(cot) theta)=\text(cos) theta$

$\text(Correct)$

Question 4 of 6

Identify if the equation is correct

$tan theta(1-cot^2theta)+cot theta(1-tan^2theta)=0$

1. $\text(Correct)$
2. $\text(Incorrect)$

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Trigonometric Identities

$\sec\theta=\frac{1}{\text{cos}\theta}$

$\csc\theta=\frac{1}{\text{sin}\theta}$

$\cot\theta=\frac{1}{\text{tan}\theta}$

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

First term

	$\text(tan) theta(1-\text(cot)^2theta)$

$=$	$\text(tan) theta-(\text(tan) theta*\text(cot)^2theta)$	Expand

$=$	$\text(tan) theta-((\text(sin) theta)/(\text(cos) theta)*\text(cot)^2theta)$	$\text(tan) theta=(\text(sin) theta)/(\text(cos) theta)$

$=$	$\text(tan) theta-((\text(sin) theta)/(\text(cos) theta)*(\text(cos)^2theta)/(\text(sin)^2theta))$	$\text(cot)^2theta=(\text(cos)^2theta)/(\text(sin)^2theta)$

$=$	$\text(tan) theta-((\text(cos) theta)/(\text(sin) theta))$	Simplify

$=$	$\text(tan) theta-\text(cot) theta$	$(\text(cos) theta)/(\text(sin) theta)=\text(cot) theta$

Second term

	$\text(cot) theta(1-\text(tan)^2theta)$

$=$	$\text(cot) theta-(\text(cot) theta*\text(tan)^2theta)$	Expand

$=$	$\text(cot) theta-((\text(cos) theta)/(\text(sin) theta)*\text(tan)^2theta)$	$\text(cot) theta=(\text(cos) theta)/(\text(sin) theta)$

$=$	$\text(cot) theta-((\text(cos) theta)/(\text(sin) theta)*(\text(sin)^2theta)/(\text(cos)^2theta))$	$\text(tan)^2theta=(\text(sin)^2theta)/(\text(cos)^2theta)$

$=$	$\text(cot) theta-((\text(sin) theta)/(\text(cos) theta))$	Simplify

$=$	$\text(cot) theta-\text(tan) theta$	$(\text(sin) theta)/(\text(cos) theta)=\text(tan) theta$

Finally, combine the two terms and add

		$\text(tan) theta-\text(cot) theta$ $+$ $\text(cot) theta-\text(tan) theta$
	$=$	$0$

This means that

$\text(tan) theta(1-\text(cot)^2theta)+\text(cot) theta(1-\text(tan)^2theta)=0$

$\text(Correct)$

Question 5 of 6

Identify if the equation is correct

$csc^4A-cot^4A=(1+cos^2A)/(sin^2A)$

1. $\text(Correct)$
2. $\text(Incorrect)$

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Mute

Trigonometric Identities

$\sec\theta=\frac{1}{\text{cos}\theta}$

$\csc\theta=\frac{1}{\text{sin}\theta}$

$\cot\theta=\frac{1}{\text{tan}\theta}$

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

	$\text(csc)^4A-\text(cot)^4A$
$=$	$(\text(csc)^2A-\text(cot)^2A)(\text(csc)^2A+\text(cot)^2A)$	Factorise
$=$	$[\text(csc)^2A-(\text(csc)^2A-1)][\text(csc)^2A+(\text(csc)^2A-1)]$	$\text(cot)^2A=\text(csc)^2A-1$
$=$	$(\text(csc)^2A-\text(csc)^2A+1)(\text(csc)^2A+\text(csc)^2A-1)$	Adjust the subtracting values
$=$	$(1)(2\text(csc)^2A-1)$	Combine like terms

$=$	$(1)(2*1/(\text(sin)^2A)-1)$	$\text(csc)^2A=1/(\text(sin)^2A)$

$=$	$(2-\text(sin)^2A)/(\text(sin)^2A)$	Apply rule of subtracting fractions

$=$	$(2-(1-\text(cos)^2A))/(\text(sin)^2A)$	$\text(sin)^2A=1-\text(cos)^2A$

$=$	$(2-1+\text(cos)^2A)/(\text(sin)^2A)$	Adjust the subtracting values

$=$	$(1+\text(cos)^2A)/(\text(sin)^2A)$

This means that

$\text(csc)^4A-\text(cot)^4A=(1+\text(cos)^2A)/(\text(sin)^2A)$

$\text(Correct)$

Question 6 of 6

Identify if the equation is correct

$(sin^2theta-tan^2theta)/(1-sec^2theta)=sin^2theta$

1. $\text(Incorrect)$
2. $\text(Correct)$

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Mute

Trigonometric Identities

$\sec\theta=\frac{1}{\text{cos}\theta}$

$\csc\theta=\frac{1}{\text{sin}\theta}$

$\cot\theta=\frac{1}{\text{tan}\theta}$

$\tan\theta=\frac{\text{sin}\theta}{\text{cos}\theta}$

$\cot\theta=\frac{\text{cos}\theta}{\text{sin}\theta}$

Work on changing the left side of the equation to make it equal to the value of the right side

	$(\text(sin)^2theta-\text(tan)^2theta)/(1-\text(sec)^2theta)$

$=$	$(\text(sin)^2theta-\text(tan)^2theta)/(-\text(tan)^2theta)$	$1-\text(sec)^2theta=-\text(tan)^2theta$

$=$	$(\text(sin)^2theta)/(-\text(tan)^2theta)-(\text(tan)^2theta)/(-\text(tan)^2theta)$	Separate the numerators

$=$	$(\text(sin)^2theta)/((\text(sin)^2theta)/(\text(cos)^2theta))+1$	$\text(tan)^2theta=(\text(sin)^2theta)/(\text(cos)^2theta)$

$=$	$-\text(cos)^2theta+1$
$=$	$1-\text(cos)^2theta$
$=$	$\text(sin)^2theta$	$1-\text(cos)^2theta=\text(sin)^2theta$

This means that

$(\text(sin)^2theta-\text(tan)^2theta)/(1-\text(sec)^2theta)=\text(sin)^2theta$

$\text(Correct)$

Proving Trigonometric Identities

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