Proving Identities 1.sec θ cot θ sin θ =1 2.cot θ • sin θ = cos θ 3.Cot θ • sec...

Questions

October 2022 1 2 Report

Proving Identities

1.sec θ cot θ sin θ =1
2.cot θ • sin θ = cos θ
3.Cot θ • sec θ = csc θ
4.(tan θ + 1 ) ( tan θ -1 )
5.(1 - cos2θ) csc2θ = 1

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KizunaHigh

Verified answer

$\large \bold {ANSWER}$

$\small\sf{ \sec(θ) \cot(θ) \sin(θ) = 1 } \\ \\ \small\textsf{By quotient and reciprocal identity} \\ \\ \small\sf{ \frac{1}{\cancel{ \cos(θ)} } \times \frac{\cancel{ \cos(θ)}}{\sin(θ)} \times \sin(θ)} \\ \\ \small\sf{ \frac{1}{\cancel{ \sin(θ)} } \times \cancel{ \sin(θ)} } \\ \\ \small\sf{ = \green1}$

$\small\sf{ \cot(θ) \sin(θ) = \cos(θ) } \\ \\ \small\textsf{By quotient identity} \\ \\\small\sf{ \frac{ \cos(θ) }{\cancel{ \sin(θ)} } \times \cancel{ \sin(θ)} } \\ \\= \green{ \cos(θ) }$

$\small\sf{ \cot(θ) \sec(θ) = \csc(θ) } \\ \\ \small\textsf{By quotient \: and \: reciprocal identity} \\ \\\small\sf{ \frac{\cancel{ \cos(θ)} }{{ \sin(θ)} } \times { \frac{1}{\cancel{ \cos(θ)} } }} \\ \\ \small\sf{ \frac{1}{\sin(θ)} = \green{ \csc(θ) }}$

$\small\textsf{I can't solve number 4, apologies.}$

$\small\sf{1 - \cos(2θ) \csc(2θ) = 1 } \\ \\ \small\textsf{By double angle identity} \\ \\ \small\sf{ \frac{ \sin(2θ) ^{2} }{2 \sin(θ) \cos(θ) } } \\ \\ \small\sf{ \frac{\cancel{2 \sin(θ) ^{2}} }{\cancel{2 \sin(θ)} \cos(θ) } } \\ \\ \small\sf{ \frac{ \sin(θ) }{ \cos(θ) } =\green{ \tan(θ) - false } }$