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Solving and Simplifying using Trig Identities

Date: 12/5/95 at 23:9:4
From: Anonymous
Subject: Trigonometry

I am starting a new chapter and there are a couple of questions that I 
can not figure out.  Could you please help?

1. Solve for x: 20.0   = 85.0

                  sec  @
2. Simplify:  1 + -----
                  cos  @

                 4       4            2
3. Show that  tan y - sec y = 1 - 2sec y

Thank you for your help.

Date: 5/30/96 at 11:2:28
From: Doctor Alex
Subject: Re: Trigonometry

Solution 1

Lg both sides. Then you will get 3x*lg20 = lg85. 
I'm sure you can solve it now:-)

Solution 2

I don't know how simple is simple :-) but since sec@=1/cos@,
sec^2@/cos^2@ = sec^4@ => 1+sec^2@/cos^2@ = 1+sec^4@ 

Solution 3

Use the very useful identity  A^2-B^2 = (A+B)(A-B)

therefore tan^4y-sec^4y = (tan^2y+sec^2y)(tan^2y-sec^2y) ----(1)

Remember the identity cos^2y + sin^2y = 1?
Divide both sides by cos^2y to get 1+tan^2y = sec^2y

Using this new identity, (tan^2y+sec^2y) becomes (2sec^2y-1) 
by adding sec^2y to both sides.

Similarly (tan^2y-sec^2y) becomes -1

Substitute into (1) and you get (2sec^2y-1) (-1) = RHS

I hope this helps. 

A^2-B^2=(A+B)(A-B) is a very very very useful identity. 
I would recommend that you remember it, because it can simplify many algebra and trig problems.

-Doctor Alex,  The Math Forum

Associated Topics:
High School Trigonometry

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