Search 75,969 tutors
FIND TUTORS
Ask a question
0 0

If tan(45+x)+tan(45-x)=3 then show tan²(45+x)+tan²(45-x)=7

Tutors, please sign in to answer this question.

1 Answer

Suggestion:
 
Since we would like to show:

If tan(45+x)+tan(45-x)=3 then show tan²(45+x)+tan²(45-x)=7
 
Let us try to square both sides of the starting equation:
tan(45+x)+tan(45-x)=3
 
Then we have:
tan2(45+x)+ 2 tan(45+x)tan(45-x) + tan2(45-x)=32
 
Two trig formulas which will be handy are:
 
tan(u + v) = (tan u + tan v) / (1 - tan u tan v)
tan(u - v) = (tan u - tan v) / (1 + tan u tan v)
 
Now let's look at the middle term:
2 tan(45+x)tan(45-x)
 
Also recall tan 45 = 1
 
By simplifying using the two trig formulas above, you will find that:
2 tan(45+x)tan(45-x) = 2
 
Then going back to the starting point we have:
 
tan2(45+x)+ 2 tan(45+x)tan(45-x) + tan2(45-x)=32
 
tan2(45+x)+ 2 + tan2(45-x)=9
 
 
Subtracting 2 from both sides we end up with we needed:
tan2(45+x)+ tan2(45-x)=7
 
Thanks,
Larry