Question

prove that (tano+coto+1)(tano+coto-1)=sec2o+cot2o​

Answer 1

SOLUTION--

TO PROVE-

(tanΦ + cotΦ + 1)(tanΦ + cotΦ - 1) = sec²Φ + cotΦ

PROOF:

The equation is of the form (a+b)(a-b)

As we know,

(a+b) (a-b) = a²-b²

So,

(tanΦ + cotΦ + 1)(tanΦ + cotΦ - 1)

= (tanΦ + cotΦ)² - (1)²

= tan²Φ + cot²Φ + 2tanΦcotΦ - 1

[(a+b)² = a² + b² + 2ab]

= tan²Φ + cot²Φ + 2 - 1

(tanΦ×cotΦ = 1)

= tan²Φ + cot²Φ + 1

= (tan²Φ + 1) + cot²Φ

= sec²Φ + cot²Φ (As, tan²Φ+1 = sec²Φ)

$\blue{\huge{Hence\:Proved}}$