If `sin2A=lambda sin2B`, then prove that `(tan(A+B))/(tan(A-B))=(lambda+1)/(lambda-1)`

If sin2A= lambda sin 2B prove that (tan(A+B)/tan(A-B))=(lambda+1)/(lambda-1)

A straight line L cuts the lines A B ,A Ca n dA D of a parallelogram A B C D at points B_1, C_1a n dD_1, respectively. If ( vec A B_1)=lambda_1 vec A B ,( vec A D_1)=lambda_2 vec A Da n d( vec A C_1)=lambda_3 vec A C , then prove that 1/(lambda_3)=1/(lambda_1)+1/(lambda_2) .

Prove that, tan((A)/(2)+B)=(c+b)/(c-b)tan((A)/(2))

Prove that (1-tan^2(pi/4-A))/(1+tan^2(pi/4-A))=sin2Adot

Prove that : tan^(-1)(1/4)+tan^(-1)(2/9)=sin^(-1)(1/sqrt5) .

If tan(A+B)=p and tan(A-B)=q , then show that tan2A=(p+q)/(1-pq) .

If 3tan A tan B=1, prove that 2cos(A+B)=cos(A-B)dot

For any triangle ABC, prove that : (sin(B-C))/(sin(B+C))=(b^(2)-c^(2))/(a^(2))

By using above basic addition/ subtraction formulae, prove that (i) tan (A+B)=(tan A+tan B)/(1-tan A tan B) , (ii) tan (A-B)=(tan A-tanB)/(1+tan A tan B) (iii) sin2theta=2sintheta costheta , (iv) cos2theta=cos^(2)theta=1-2sin^(2)theta=2cos^(2)theta-1 (v) tan 2theta=(2 tan theta)/(1-tan^(2) theta)

Prove that sec A (1 - sin A)(sec A+ tan A) = 1.