Question

Each of the following expressions has a single numerical value for all θ where the expression is defined. Determine the numerical value of each expression and make sure to enter a single number.√36⋅cos^2(θ)+36⋅sin^2(θ)=7/sec^2(θ)−tan^2(θ)=(cos^2(θ)+sin^2(θ))(sec^2(θ)−tan^2(θ))=sin(θ)/csc(θ)+cos(θ)/sec(θ)=

Answer 1

Part a

Remember that

`\sin ^2(\theta)+\cos ^2(\theta)=1`

therefore

`\begin{gathered} \sqrt[]{36\sin^2(\theta)+36\cos^2(\theta)} \\ \sqrt[]{36(\sin^2(\theta)+\cos^2(\theta))} \\ \sqrt[]{36} \\ 6 \end{gathered}`

Part b

Remember that

`\begin{gathered} \tan ^2(\theta)+1=\sec ^2(\theta) \\ \end{gathered}`

substitute in the given expression

`\begin{gathered} (7)/(\sec^2(\theta)-\tan^2(\rbrack\theta)) \\ \\ (7)/(\tan ^2(\theta)+1-\tan ^2(\theta)) \\ \\ (7)/(1)=7 \end{gathered}`

Part c

Substitute the given identities in part a and part b

we have

`(1)\cdot(1)=1`

Part d

Remember that

`\csc (\theta)=(1)/(\sin (\theta))`
`\sec (\theta)=(1)/(\cos (\theta))`

substitute in the given expression

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