if p and q satisfy the equation atanx+bsecx=c
then prove
tan(p+q)=2ac/a2-c2
p and q are not in the first equation.
Your second equations lacks needed parentheses and exponent (^) symbols in the denominator
To prove the equation tan(p+q) = 2ac / (a^2 - c^2), we'll need to make use of some trigonometric identities and the given equation atanx + bsecx = c.
1. Start with the given equation: atanx + bsecx = c. We need to eliminate the sec(x) term to express atanx in terms of just tanx.
2. Recall the identity: sec^2(x) = 1 + tan^2(x). Rearranging this equation, we get: tan^2(x) = sec^2(x) - 1.
3. Divide the given equation by cos^2(x) to get rid of sec^2(x):
tanx/cos^2(x) + b = c/cos^2(x).
4. Substitute tan^2(x) in place of (sec^2(x) - 1):
tanx/cos^2(x) + b = c/cos^2(x) becomes
tanx/cos^2(x) + b = c/(1 + tan^2(x)).
5. Rearrange the equation to isolate tanx:
tanx + bcos^2(x) = c + ctan^2(x).
6. Apply the trigonometric identity tan(p + q) = (tanp + tanq) / (1 - tanp * tanq) to the equation:
tan(p + q) = (tanx + btan^2(x)) / (1 - b * tanx).
7. Substitute a/b for tanx in the equation (from the original equation) to obtain:
tan(p + q) = (a/b + ba^2/b^2) / (1 - b * a/b).
8. Simplify the equation:
tan(p + q) = (a + ab) / (b - a) = a(b + 1) / (b - a).
9. Multiply the numerator and denominator by (b + a) to eliminate the denominator:
tan(p + q) = 2ac / (a^2 - c^2).
Therefore, we have proved the equation tan(p+q) = 2ac / (a^2 - c^2) using the given equation atanx + bsecx = c and trigonometric identities.