Find the value of (sin 30° + tan 45° - cosec 60°) / (sec 30° + cos 60° + cot 45°) and (5 cos² 60° + 4 sec² 30° - tan² 45°) / (sin² 30° + cos² 30°) | Trigonometry Numerical

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Question 1: Find the value of sin30°+ tan45° cosec60°sec30°+ cos60°+cot45°{\sin 30 \degree + \ \tan 45 \degree - \ \cosec 60 \degree} \over {\sec 30 \degree + \ \cos 60 \degree + \cot 45 \degree}

sin30°+ tan45° cosec60°sec30°+ cos60°+cot45°=12+12323+12+1=322323+32=3342333+423(taking LCM in numerator and denominator)=3342333+423=(334)(33+4)=(334)(334)(33+4)(334)rationalizing denominator=(334)2(33)2(4)2using (a+b)(ab)=a2b2 formula for denominator=(33)22(33)4+422716=27243+1611=4324311\begin{aligned} &{\sin 30 \degree + \ \tan 45 \degree - \ \cosec 60 \degree} \over {\sec 30 \degree + \ \cos 60 \degree + \cot 45 \degree} \\ \\ = &{{1 \over 2} + 1 - {2 \over \sqrt3} \over {2 \over \sqrt3} + {1 \over 2} + 1} \\ \\ = &{{3 \over 2} - {2 \over \sqrt3} \over {2 \over \sqrt3} + {3 \over 2}} \\ \\ = &{{3 \sqrt3 - 4 \over 2 \sqrt3} \over {3 \sqrt3 + 4 \over 2 \sqrt3}} \quad --- \text{(taking LCM in numerator and denominator)} \\ \\ = &{{3 \sqrt3 - 4 \over \cancel{2 \sqrt3}} \over {3 \sqrt3 + 4 \over \cancel{2 \sqrt3}}} \\ \\ = &{\big(3 \sqrt3 - 4 \big) \over \big( 3 \sqrt3 + 4 \big)} \\ \\ = &{\big(3 \sqrt3 - 4 \big)\big(3 \sqrt3 - 4 \big) \over \big( 3 \sqrt3 + 4 \big)\big(3 \sqrt3 - 4 \big)} \quad --- \text{rationalizing denominator} \\ \\ = &{\big(3 \sqrt3 - 4 \big)^2 \over (3 \sqrt3)^2 - (4)^2} \quad --- using \ (a +b)(a-b) = a^2 - b^2 \ \text{formula for denominator} \\ \\ = &{(3 \sqrt3)^2 - 2 \sdot(3 \sqrt3) \sdot 4 + 4^2 \over 27 - 16} \\ \\ = &{27 - 24 \sqrt3 + 16 \over 11} \\ \\ &\boxed{={43 - 24 \sqrt3 \over 11}} \end{aligned}

Question 2: Find the value of 5cos260°+4sec230°tan245°sin230°+cos230°{5 \cos ^2 60 \degree + 4 \sec^2 30 \degree - \tan^2 45 \degree \over \sin^2 30 \degree + \cos^2 30 \degree}

5cos260°+4sec230°tan245°sin230°+cos230°=5(12)2+4(23)2(1)2(12)2+(32)2=5(14)+4(43)114+34=54+163144=15+6412121taking LCM in numerator and denominator=6712\begin{aligned} &{5 \cos ^2 60 \degree + 4 \sec^2 30 \degree - \tan^2 45 \degree \over \sin^2 30 \degree + \cos^2 30 \degree} \\ \\ = &{5\big({1 \over 2} \big)^2 + 4 \big( {2 \over \sqrt3}\big)^2 - (1)^2 \over \big({1 \over 2} \big)^2 + \big( {\sqrt3 \over 2}\big)^2} \\ \\ = &{5\big({1 \over 4} \big) + 4 \big( {4 \over 3}\big) - 1 \over {1 \over 4} + {3 \over 4}} \\ \\ = &{{5 \over 4} + {16 \over 3} - 1 \over {4 \over 4}} \\ \\ = &{{15 + 64 - 12 \over 12} \over 1} \quad --- \text{taking LCM in numerator and denominator}\\ \\ &\boxed{={67 \over 12}} \end{aligned}

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