Question

NO LINKS!! Verify each identity. Show work please



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Answer 1

#33

LHS

`\\ \rm\Rrightarrow (sin\theta)/(csc\theta)+(cos\theta)/(sec\theta)`

`\\ \rm\Rrightarrow (sin\theta)/((1)/(sin\theta))+(cos\theta)/((1)/(cos\theta))`

`\\ \rm\Rrightarrow sin^2\theta+cos^2\theta`

`\\ \rm\Rrightarrow 1`

• Proved

#34

• Ø is taken as A for easy typing

LHS

`\\ \rm\Rrightarrow tanAcsc^2A-tanA`

`\\ \rm\Rrightarrow (sinA)/(cosA)* (1)/(sin^2A)-(sinA)/(cosA)`

`\\ \rm\Rrightarrow (sinA)/(sin^2AcosA)-(sinA)/(cosA)`

`\\ \rm\Rrightarrow (1)/(sinAcosA)-(sinA)/(cosA)`

`\\ \rm\Rrightarrow (1-sin^2A)/(sinAcosA)`

`\\ \rm\Rrightarrow (cos^2A)/(sinAcosA)`

`\\ \rm\Rrightarrow (cosA)/(sinA)`

`\\ \rm\Rrightarrow cotA`

Proved

Answer 2

Trigonometric Identities used:

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

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

`\cot \theta=(1)/(\tan \theta)`

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

`\csc^2 \theta = 1 + \cot^2 \theta`

Question 33

`\large \begin{aligned}(\sin \theta)/(\csc \theta)+(\cos \theta)/(\sec \theta)& = \sin \theta \cdot (1)/(\csc \theta)+\cos \theta \cdot (1)/(\sec \theta)\\\\& = \sin \theta \cdot (1)/((1)/(\sin \theta))+\cos \theta \cdot (1)/((1)/(\cos \theta))\\\\& = \sin \theta \cdot \sin \theta+\cos \theta \cdot \cos \theta\\\\& = \sin^2 \theta + \cos^2 \theta\\\\& = 1\end{aligned}`

Question 34

`\large \begin{aligned}\tan \theta \csc^2 \theta - \tan \theta & = \tan \theta ( \csc^2 \theta - 1)\\\\& = \tan \theta (1 + \cot^2 \theta -1)\\\\& = \tan \theta \cot^2 \theta \\\\& = (1)/(\cot \theta) \cdot \cot^2 \theta \\\\& = (\cot^2 \theta)/(\cot \theta)\\\\& = \cot \theta\end{aligned}`