Question 1170774
1.sin(θ) = cos(θ) tan(θ)

Start with tan(θ) = sin(θ)/cos(θ), then multiply both sides by cos(θ)


2. 1/cos(θ)=tan(θ)/sin(θ)

Start with tan(θ) = sin(θ)/cos(θ), then divide both sides by sin(θ)<br>

3. {{{cos^2(theta) tan^2(theta) = sin^2(theta) }}}

Starting with
tan(θ) = sin(θ)/cos(θ)  

we get:
 {{{ tan^2(theta) = sin^2(theta)/cos^2(theta) }}}

Next just multiply both sides by {{{ cos^2(theta) }}}
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4. sin(θ)tan(θ) + cos(θ) = 1/cos(θ)

Using
tan(θ) = sin(θ)/cos(θ)  

Re-write LHS of given equation (#4) as
= sin(θ)*sin(θ)/cos(θ) + cos(θ)

Put both terms over cos(θ):
= {{{ sin^2(theta)/cos(theta) + cos^2(theta)/cos(theta) }}}<br>
= {{{ (sin^2(theta) + cos^2(theta)) / cos(theta) }}}<br>

Noting that {{{ (sin^2(theta) + cos^2(theta)) = 1 }}}:<br>

= {{{ 1/cos(theta) }}}

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I leave #5 for you to do, there are enough examples here.  As a hint: start with the LHS, multiply thru by sin(θ) then factor out tan(θ).