Question

if cosec theta + cot theta = p the prove that cos theta= p square-1/p square+1

Answer 1

p^2 - 1 = 2 cosec^2 theta + 2 cosec theta x cos theta

P sq +1= 2 cosec squared theta + 2 cos theta x cos theta
p sq - 1 / p^ sq +1 = cosec theta - cot theta / cosec theta + cot theta

break in this form

cosec theta = 1/ sin
cot theta = cos/ sin

after that

1 - cos/ 1+ cos

multiply by 1 + cos numerator and denominator
.and finally we get
= cos theta

Answer 2

Cosec A + Cot A = P

= 1/Sin A + Cos A / Sin A = P

= 1+Cos A/Sin A = p

=> SQUARING ON BOTH SIDES

= (1+COS A)²/(SINA)²= P²

= (1+ Cos A)² / (Sin A)² = P²

= (1+cos A)² = (p²)[(Sin A)²]

= (1+ cos A) ² = (p²) [(1-cos²A)]

= (1+ cos A) ² = (p²) [ (1+cosA)(1-cos A) ]

= (1+cos A)² ÷ (1+ cos A) = (p²)[(1-cos A)]

= 1+cos A = (p²)[1-cos A]

= 1+cos A ÷ 1-cos A = p²

= Here, Using (a+b/a-b=c+d/c-d). This is known as Componendo and dividendo

According to the Question statement!


1+cos A ÷ 1-cos A = p²

Then,

(1+cos A) + (1 - cos A ) ÷ (1+ Sin A - (1-sin A) = p²+1 / p²-1

= 2/2cos = p²+1/p²-1

= 1/cos = p²+1/p²-1

= Sec = p²+1 /p²-1

We know that Cos A = 1/sec A

Then,

Cos A = p²-1 /p²+1