Prove That. cos2 30° + sin 30° + tan2 45° = 2 ¼

Prove That.

(i) cos² 30° + sin 30° + tan² 45° = 2 ¼

(ii) 4 (sin⁴ 30° + cos⁴ 60°) – 3 (cos² 45° – sin² 90°) = 2

(iii) cos 60° = cos² 30° – sin² 30°.

Answer :

(i) cos² 30° + sin 30° + tan² 45° = 2 ¼

LHS = cos² 30° + sin 30° + tan² 45°

= (√3/2)² + ½ + 1²

= ¾ + ½ + 1

= (3 + 2 + 4)/4

= 9/4

= 2 ¼

= RHS

Hence, LHS = RHS.

(ii) 4 (sin⁴ 30° + cos⁴ 60°) – 3 (cos² 45° – sin² 90°) = 2

LHS = 4 (sin⁴ 30° + cos⁴ 60°) – 3 (cos² 45° – sin² 90°)

= 4[(½)⁴ + (½)⁴] – 3 [(1/√2)² – 1²]

= 4[½ × ½ × ½ × ½ + ½ × ½ × ½ × ½] – 3 [½ – 1]

= 4 [1/16 + 1/16] – 3 (- ½)

= 4[(1 + 1)/16] + 3/2

= (4 × 3)/16 + 3/2

= 8/16 + 3/2

= ½ + 3/2

= (1 + 3)/2

= 4/2

= 2

= RHS

Hence, LHS = RHS.

(iii) cos 60° = cos² 30° – sin² 30°

LHS = cos 60° = ½

RHS = cos² 30° – sin² 30°

= (√3/2)² + (1/2)²

= ¾ – ¼

= (3 – 1)/4

= 2/4

= ½

= RHS

Hence, LHS = RHS.

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