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Q3. Evaluate :

(i) \( \frac{sin^263° + sin^227°}{cos^217° + cos^273°} \)

(ii) \( sin 25° cos 65° \ + \ cos25° sin65° \)

Answer :


(i) \( \frac{sin^263° + sin^227°}{cos^217° + cos^273°} \ = \ \frac{cos^227°+sin^227°}{ sin^273° + cos^273°} \)

[ ∵ \( sin(90° - \theta) \ = \ cos\theta \) => \( sin63° \ = \ sin(90° – 27°) \ = \ cos27° \) ]

and [∵ \( cos(90° - \theta) \ = \ sin\theta \)] => \( cos17° \ = \ cos(90° – 73°) \ = \ sin 73° \) ]

\( = \ \frac{1}{1} \ = \ 1 \)    [∵ \( cos^2A + sin^2A \ = \ 1 \) ]

(ii) \( sin25° cos65° + cos25°sin65° \)

= \( sin(90°– 65°). cos65° + cos(90°– 65°) sin65° \)

= \( cos65° cos65° + sin 65° sin65° \)    [∵ \( sin(90°-\theta ) \ = \ cos\theta \) and \( cos(90°-\theta ) \ = \ sin\theta \) ]

= \( cos^265° + sin^265° \ = \ 1 \)