trigo quadratic

It is given that tan A and tan B are the roots of t²-pt+q=0. Find in terms of p and q:

i) sin²(A+B)
ii)cos 2(A+B)

2 Answers

sin ² (A+B) = p² / {p² + (1 - q )²}

cos 2(A+B) = { (1-q)² - p² } / { (1 - q)² + p² }

you have to express sin²(A+B) in terms of tan A and tan B

Which is not very difficult if you use

sec²(A+B) = 1+ tan²(A+B) = 1+\left[\frac{tan A+ tan B}{1-tan A \times tan B} \right]^2

You know tan A + tan B = p

and tan A. tan B =q

Now solve :)

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