Prove the following:- If x = asecA + btanA and y = atanA + bsecA, prove that x2 - - Maths - Introduction to Trigonometry

Question

Prove the following:-

If x = asecA + btanA and y = atanA + bsecA, prove that
x2 - y2 = a2 - b2
Whole Root 1 + cosA / 1 - cosA + Whole Root 1 - cosA / 1 + cosA
= 2cosecA

Abhishek Kr. Jha · Accepted Answer

1) x = a secA+b tanA; y = atanA+bsecASo we have;L H S = x2-y2= a
secA+b tanA2-atanA+bsecA2= a2sec2A+b2tan2A+2ab secA
tanA-a2tan2A+b2sec2A+2ab secA tanA=
a2sec2A+b2tan2A-a2tan2A-b2sec2A= a2sec2A-tan2A-b2sec2A-tan2A= a2-b2
{Since; sec2A-tan2A = 1}= R H S2) L H S = 1+cosA1-cosA+
1-cosA1+cosA= 1+cosA1-cosA×1+cosA1+cosA+
1-cosA1+cosA×1-cosA1-cosA= 1+cosA212-cos2A+1-cosA212-cos2A=
1+cosAsin2A+1-cosAsin2A= 1+cosAsinA+1-cosAsinA= 1+cosA+1-cosAsinA=
2sinA= 2cosecA= R H S

Prove the following:- If x = asecA + btanA and y = atanA + bsecA, prove that x2 - y2 = a2 - b2 Whole Root 1 + cosA / 1 - cosA + Whole Root 1 - cosA / 1 + cosA = 2cosecA.

Prove the following:-

If x = asecA + btanA and y = atanA + bsecA, prove that x² - y² = a² - b²

Whole Root 1 + cosA / 1 - cosA + Whole Root 1 - cosA / 1 + cosA = 2cosecA.