If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m 2 - n 2 = a 2 - b 2

To prove: m 2 - n 2 = a 2 - b 2 Substituting value of m and n in L.H.S. of the above equation : = (a sec A + b tan A) 2 - (a tan A + b sec A) 2 = a 2 sec 2 A + b 2 tan 2 A + 2ab sec A tan A - (a 2 tan 2 A + b 2 sec 2 A + 2ab sec A tan A) = a 2 sec 2 A - a 2 tan 2 A + b 2 tan 2 A - b 2 sec 2 A + 2ab sec A tan A - 2ab sec A tan A = a 2 (sec 2 A - tan 2 A) + b 2 (tan 2 A - sec 2 A) = a 2 (sec 2 A - tan 2 A) - b 2 (sec 2 A - tan 2 A) By formula, sec 2 A - tan 2 A = 1 ⇒ a 2 × 1 - b 2 × 1 ⇒ a 2 - b 2 Since, L.H.S. = R.H.S. Hence, proved that m 2 - n 2 = a 2 - b 2 .

Mathematics

If m = a sec A + b tan A and n = a tan A + b sec A, then prove that :
m² - n² = a² - b²

Trigonometric Identities

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Answer

To prove:

m² - n² = a² - b²

Substituting value of m and n in L.H.S. of the above equation :

= (a sec A + b tan A)² - (a tan A + b sec A)²

= a² sec² A + b² tan² A + 2ab sec A tan A - (a² tan² A + b² sec² A + 2ab sec A tan A)

= a² sec² A - a² tan² A + b²tan² A - b² sec² A + 2ab sec A tan A - 2ab sec A tan A

= a² (sec² A - tan² A) + b² (tan² A - sec² A)

= a² (sec² A - tan² A) - b² (sec² A - tan² A)

By formula,

sec² A - tan² A = 1

⇒ a² × 1 - b² × 1

⇒ a² - b²

Since, L.H.S. = R.H.S.

Hence, proved that m² - n² = a² - b².

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Mathematics

If m = a sec A + b tan A and n = a tan A + b sec A, then prove that :
m² - n² = a² - b²

Trigonometric Identities

Answer

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