Question

Evaluate the following.
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$\tt{1) \: \: \frac{cos \: 45°}{sec \: 30°+ \: cosec \: 30°}}$
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$\tt{2)\frac{sin \: 30° \: + \: tan \: 45°- \: cosec \: 60°}{sec \: 30°+ \: cos \: 60°+ \: cot \: 45°}}$
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Answer 1

Inorder to solve these type of questions, you must learn the value of trigonometric ratios of different angles.

Another concept used in the solution is the concept of rationalising where we multiply both numerator and denominator with the conjugate.

Below is the complete table which is going to be used in this question.

${ \begin{array}{|c|c|c|c|c|c|} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \\\sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } &\dfrac{ \sqrt{3}}{2}& 1 \\ \\\cos & 1 & \dfrac{ \sqrt{ 3 }}{{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \\ \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \\ \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \\\sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \\ \rm cosec & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \end{array}}$

Required solution is in the attachment just for the purpose of saving time.

Identity used :-

• ( A - B )² = A² + B² - 2AB
• ( A + B ) ( A - B ) = A² - B²