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Q1. Evaluate :
(i) \( sin 60 ° cos 30° + sin 30° cos 60° \)

(ii) \( 2tan^245° + cos^230° - sin^260° \)

(iii) \( \frac{cos45°}{sec30° + cosec30°} \)

(iv) \( \frac{sin30° + tan45°- cosec60°}{sec30° +cos60° +cot45°} \)

(v) \( \frac{5cos^260° +4sec^230° - tan^245°}{sin^230° + cos^230°} \)

Answer :


(i) \( sin 60° cos 30° \ + \ sin 30° cos 60 \ = \ \frac{ \sqrt{3}}{2} × \frac{ \sqrt{3}}{2} \ + \ \frac{1}{2} × \frac{1}{2} \ = \ \frac{3}{4} \ + \ \frac{1}{4} \ = \ \frac{3+1}{4} \ = \ 1 \)



(ii)\( 2tan^245° \ + \ cos^230° \ - \ sin^260° \ = \ 2(1)^2 \ + \ ( \frac{ \sqrt{3}}{2})^2 \ - \ ( \frac{ \sqrt{3}}{2})^2 \ = \ 2 \ + \ \frac{3}{4} \ - \ \frac{3}{4} \ = \ 2 \)



(iii) \( \frac{cos45°}{sec30° + cosec30°} \ = \ \frac{ \frac{1}{ \sqrt{2}}}{ \frac{2}{ \sqrt{3}} + 2} \ = \ \frac{ \frac{1}{ \sqrt{2}}}{ \frac{2 + 2 \sqrt{3}}{ \sqrt{3}}} \ = \ \frac{1}{ \sqrt{2}} × \frac{ \sqrt{3}}{2+2 \sqrt{3}} \ = \ \frac{ \sqrt{3}}{ \sqrt{2} × 2( \sqrt{3} + 1)} × \frac{ \sqrt{3} - 1}{ \sqrt{3} - 1} \ = \ \frac{ \sqrt{3}( \sqrt{3} - 1)}{ \sqrt{2} × 2(3 - 1)} \ = \ \frac{ \sqrt{2} × \sqrt{3}( \sqrt{3} - 1)}{ \sqrt{2} × \sqrt{2} × 2 × 2} \ = \ \frac{3 \sqrt{2} - \sqrt{6}}{8} \)



(iv) \( \frac{sin30° + tan45°- cosec60°}{sec30° +cos60° +cot45°} \) \( = \ \frac{ \frac{1}{2} + 1 - \frac{2}{ \sqrt{3}}}{ \frac{2}{ \sqrt{3}} + \frac{1}{2} + 1} \) \( = \ \frac{ \frac{1 + 2}{2} - \frac{2}{ \sqrt{3}}}{ \frac{2}{ \sqrt{3}} + \frac{1 + 2}{2}} \) \( = \ \frac{ \frac{3}{2} - \frac{2}{ \sqrt{3}}}{ \frac{2}{ \sqrt{3}} + \frac{3}{2}} \) \( = \ \frac{ \frac{3 \sqrt{3} - 4}{2 \sqrt{3}}}{ \frac{4 + 3 \sqrt{3}}{2 \sqrt{3}}} \) \( = \ \frac{3 \sqrt{3} - 4}{4 +3 \sqrt{3}} \) \( = \ \frac{(3 \sqrt{3} - 4)(3 \sqrt{3} - 4)}{(4 + 3 \sqrt{3})(3 \sqrt{3} - 4)} \) \( = \ \frac{27 + 16 - 12 \sqrt{3} - 12 \sqrt{3}}{27- 16} \) \( = \ \frac{43 - 24 \sqrt{3}}{11} \)



(v) \( \frac{5cos^260° +4sec^230° - tan^245°}{sin^230° + cos^230°} \) \( = \ \frac{5( \frac{1}{2})^2 + 4( \frac{2}{ \sqrt{3}})^2 - (1)^2}{( \frac{1}{2})^2 + ( \frac{ \sqrt{3}}{2})^2} \) \( = \ \frac{5 × \frac{1}{4} + 4 × \frac{4}{3} - 1}{ \frac{1}{4} + \frac{3}{4}} \) \( = \ \frac{5}{4} \ + \ \frac{16}{3} - 1 \) \( = \ \frac{15 + 64 - 12}{12} \ = \ \frac{67}{12} \)