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

Here x is theta,and please any one help me I am not getting it

### 1 Answer by Expert Tutors

Larry M. | Mathematics TutorMathematics Tutor
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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