Question

tanA/(1+tan² A)²+cotA/(1+cot² A)²= sin A cos A ,Prove it

Answer 1

Important formulas:

(1) Sin^2 A + cos^2 A = 1

(2) cot A = cos A/sin A

(3) 1/sin^2 A = cosec^2 A

(4) 1/Cos^2A = sec^2 A

(5) 1 + cot^2 A = cosec^2 A

(6) 1 + tan^2 A = sec^2 A



The answer is explained in the attachment.


Hope it helps!

Answer 2

$\bf{LHS}$ = tanA/(1 + tan²A)² + cotA/(1 + cot²A)²
we know, sec²Ф - tan²Ф = 1 ⇒ 1 + tan²Ф = sec²Ф
Similarly, cosec²Ф - cot²Ф = 1 ⇒ 1 + cot²Ф = cosec²Ф
∴ (1 + tan²A) = sec²A and (1 + cot²A) = cosec²A

= tanA/(sec²A)² + cotA/(cosec²A)²
= tanA/sec⁴A + cotA/cosec⁴A
= tanA.cos⁴A + cotA.sin⁴A
= sinA/cosA.cos⁴A + cosA/sinA.sin⁴A
= sinA.cos³A + cosA.sin³A
= sinA.cosA(cos²A + cos²A)

We know, sin²Ф + cos²Ф = 1 so, cos²A + sin²A = 1
= sinA.cosA(cos²A + sin²A) = sinA.cosA.1 = sinA.cosA =$\bf{ RHS}$