Question

TRIGONOMETRY<br /><br />
Find the value of the following(11-16<br />
)<br />
11. <br />
12. Sec 500 Sin 400 + Cos 400 Cosec 500<br />
13. tan 350 tan 400 tan 450 tan 500 tan 550 <br /><br /><br />
14 Tan 70 tan 230 tan 600 tan 670 tan 830 <br />
15. <br />
16. tan 50 tan 250 tan 300 tan 650 tan 850<br />
17 If A and B are acute angles and sin A = cos B, Prove that A + B = 900. <br />
18 . If A, B, C are the interior angles of a triangle ABC, show that show<br />
that <br />
19 Express the following in terms of t-ratios of angles between 0° and 45°. <br /><br />
1) sin 85° +cosec 85° <br /><br />
2) cosec 69° +cot 69° <br /><br />
3) sin 81° +tan 81° <br /><br /><br />
20 Prove the trigonometric identities (sec4A – sec2A) = tan4A +tan2A = sec 2 A tan2 A <br /><br />
21 Prove the trigonometric identities cotA – tanA = (2cos 2A -1)/ (sinA.cosA) <br /><br />
22 Prove the trigonometric identities. (1- sinA +cosA)2 = 2(1+cosA )(1 – sinA) <br /><br />
23 If tanA +sinA = m and tanA – sinA=n show that m2 – n2 = 4 <br /><br />
24 If x= psecA + qtanA and y= ptan A +q secA prove that x2 – y2 = p2 – q2 <br /><br />
25. If sinA + sin2A = 1 prove that cos2 A + cos4 A =1<br />
26. Prove that 2(sin6A+ cos6A) -3(sin4A+cos4A)+ 1 =0<br />
27. Prove that sinA(1+tanA) + cosA(1+cotA) = secA +cosecA<br />
28. If sinA + 2cosA = 1 , then prove that 2sinA- cosA =2<br />
29 If acosA –bsinA =x and asinA+bcosA = y Prove that a2 + b2 =x2+y2<br />
30. If cosecA +cotA =m and cosecA – co<br />

Answer 1

12.  sec 50° sin 40° + cos 40° cosec 50°
     = sin 40 / cos 50  + cos 40 / sin 50
      = [ sin 40 sin 50 + cos 40 cos 50 ] / [sin 50 cos 50]
      = cos (50 - 40) /  [(sin 100)/2]
      = 2 cos 10 / cos 10
      = 2

13.   tan 35° tan 40°  tan 45°  tan 50°  tan 55°
        tan 35° tan 40°  1    cot(90-50)°   cot(90-55)°
        tan 35   tan 40   cot 40  cot 35
        = 1

14.   tan 7°  tan 23°  tan 60°   tan 67°  tan 83°
        tan 7    tan 23   √3          cot (90-67)°   cot (90-83)°
        tan 7   tan 23    √3     cot 23  cot 7
        = √3

16.   tan 5°  tan 25°  tan 30°  tan 65°  tan 85°
        tan 5°  tan 25°   1/√3      cot (90-65)°  cot (90-85)
        = 1/√3