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3. sec 4A = cosec (A-20)
→ 1/cos 4A = 1/sin (A-20)
→ sin (A-20) = cos 4A
→ sin (A-20) = sin (90-4A) [cos x = sin (90-x)]
→ A - 20 = 90 - 4A
→ 5A = 110
→ A = 22°
4. To prove- cotA + tanA = secA . cosecA
L.H.S.- cotA + tanA = 1/tanA + tanA
→ 1+tan²A/tanA
→ sec²A/tanA
→ sec²A/sinA.secA
→ secA/sinA = secA . cosecA = R.H.S
HENCE PROVED
5. a cosx - b sinx = x
a sinx + b cosx = y
x²+y² = a²cos²x + b²sin²x - 2abcosxsinx + a²sin²x + b²cos²x + 2absinxcosx
→ x²+y² = a²(cos²x + sin²x) + b²(cos²x + sin²x)
→ x²+y² = a²+b² [cos²x+sin²x=1]
6. L.H.S.- (1 + tan²x) + (1 + 1/tan²x)
→ (sec²x) + (1 + cot²x) [1+tan²x=sec²x]
→ sec²x + cosec²x [1+cot²x=cosec²x]
→ 1/cos²x + 1/sin²x
→ sin²x+cos²x/sin²x cos²x
→ 1/sin²x(1-sin²x)
→ 1/sin²x - sin⁴x
HENCE PROVED
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