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Can anyone help me with these? I don’t think I did them correctly

Can anyone help me with these? I don’t think I did them correctly-example-1
User TechRemarker
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1 Answer

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12 votes

Answer:

1.
sec(x)-1

2.
2 sec^2(x)

3.
sec(x)-1

Explanation:

1.
tan(x)[csc(x) - cot(x)]

expand the brackets:


\implies tan(x)csc(x)-tan(x)cot(x)

Substitute the following equivalents:


tan(x)=(sin(x))/(cos(x)),
csc(x)=(1)/(sin(x)),
cot(x)=(1)/(tan(x))


\implies (sin(x))/(cos(x)) \cdot (1)/(sin(x))-(tan(x))/(1) \cdot (1)/(tan(x))

Cancel the common factors:


\implies (1)/(cos(x))-1


\implies sec(x)-1

2.
(1)/(sin(x)+1)-(1)/(sin(x)-1)


\implies ((sin(x)-1)-(sin(x)+1))/((sin(x)+1)(sin(x)-1))


\implies (-2)/(sin^2(x)-1)

Using identity
sin^2(x)+cos^2(x)=1


\implies (-2)/(-cos^2(x))


\implies (2)/(cos^2(x))


\implies 2 sec^2(x)

3.
(tan^2(x))/(sec(x)+1)

Using identity
1 + tan^2(x)=sec^2(x):


\implies (sec^2(x)-1)/(sec(x)+1)


\implies ((sec(x)+1)(sec(x)-1))/(sec(x)+1)

Cancel common factor
sec(x)+1:


\implies sec(x)-1

User Adam Fischer
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