Question

Find the value of-

cos50° / 2sin40° + 4(cosec^2 59-tan^2 31° ) / 3 tan^2 45° - 2/3 tan 12° tan 78° sin 90°

Answer 1

Thank you for your question

Answer 2

$Value \:of \: \frac{cos 50\degree}{2sin 40\degree} + \frac{4(cosec^{2}59-tan^{2} 31)}{3tan^{2} 45\degree} - \frac{2}{3} tan 12\degree tan 78 \degree sin 90 \degree$

$= \frac{cos (90-40\degree)}{2sin 40\degree} + \frac{4(cosec^{2}(90-31)-tan^{2} 31)}{3\times 1} - \frac{2}{3} tan 12\degree tan(90-12 \degree )\times 1$

$= \frac{sin 40\degree}{2sin 40\degree} + \frac{4(sec^{2}31-tan^{2} 31)}{3} - \frac{2}{3} tan 12\degree cot12 \degree$

$= \frac{1}{2} + \frac{4\times 1}{3} - \frac{2}{3} \times 1$

____________________

/* By Trigonometric Identity */

• Sec² A - tan²A = 1
• tanAcotA = 1

____________________

$= \frac{1}{2} + \frac{4}{3} - \frac{2}{3} \\= \frac{ 3+8-4}{6} \\= \frac{7}{6}$

Therefore.,

$\red{Value \:of \: \frac{cos 50\degree}{2sin 40\degree} + \frac{4(cosec^{2}59-tan^{2} 31)}{3tan^{2} 45\degree} - \frac{2}{3} tan 12\degree tan 78 \degree sin 90 \degree}$

$\green {= \frac{7}{6}}$

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