Prove: - sec 4A(1 – sin 4A) – 2 tan 2A = 1.

Formula used: -

(i) sec²A – tan²A = 1

(ii)

(iii)

(iv) (a² – b²) = (a + b)(a – b)

taking L.H.S

Sec⁴A(1 – sin⁴A) – 2tan²A = sec⁴A – sec⁴Asin⁴A – 2tan²A

using formula (ii)

using formula (iii)

⇒ Sec⁴A(1 – sin⁴A) – 2tan²A = ((sec²A)² – (tan²A)²) – 2tan²A

using formula (iv)

⇒ Sec⁴A(1 – sin⁴A) – 2tan²A = (sec²A + tan²A)(sec²A – tan²A) – 2tan²A

using formula (i)

⇒ Sec⁴A(1 – sin⁴A) – 2tan²A = sec²A + tan²A – 2tan²A = sec²A – tan²A

⇒ Sec⁴A(1 – sin⁴A) – 2tan²A = 1