Question

Cot²Φ/CosecΦ—1=1+SinΦ/1—SinΦ​

Answer 1

Correct Question

Prove that cot²∅/(cosec∅ - 1) = (1 + sin∅)/sin∅.

Proof

L.H.S → cot²∅/(cosec∅ - 1)

L.H.S → cot²∅/(1/sin∅ - 1)

L.H.S → (cos∅/sin∅)²/(1 - sin∅)/sin∅

L.H.S → cos²∅/sin²∅ × sin∅/(1 - sin∅)

L.H.S → cos²∅/sin∅ × 1/(1 - sin∅)

L.H.S → (1 - sin²∅)/sin∅(1 - sin∅)

L.H.S → (1 + sin∅)(1 - sin∅)/sin∅(1 - sin∅)

L.H.S → (1 + sin∅)/sin∅ = R.H.S

Remember

• 1 - sin²∅ = cos²∅
• cot∅ = cos∅/sin∅
• cosec∅ = 1/sin∅
• 1 - sin²∅ = (1 + sin∅)(1 + sin∅) due to a² - b² identity.

Hence Proved