Question

solve questions from 4 to 14
50 points

Answer 1

14.
tan x+cot x=4
(tan x+cot x)^2=16
tan^2 x + cot^2 x+2=16
tan^2 x+cot^2 =14
(tan^2 x+cot^2 x)^2=196
tan^4 x+cot^4 x+2=196
ans=194
5.
sin x+cosec x=2
cosec x=1/sin x
x=90°
sin^n 90°+(1/sin^n 90°)=2
sin 90°=1
1 power all is 1
8.
theta 1= theta 2= theta 3=90°
cos 90°=0
ans=0
9.
sin(360°-x)= -sin x
sin 10°+sin 20°+.........sin180°+sin(360°-170°)+sin(360°-160°)+......sin(360°-10°)
all are cancel so
ans 0
12.
cos(180°-179°)+ cos(180°-178°)+.....+cos178°+cos 179°+cos180°
cos(180°-179°)= -cos 179°
all are cancel but cos 180°=-1
ans:-1
6.sin x+sin^2 x=1
sin x=1-sin^2 x
sin x=cos^2 x
cos^12 x+3cos^10 x+3 cos^8 x+cos^6 x
cos^12 x=sin^6 x
cos^10 x=sin^5x
cos^8 x=sin^4 x
cos^6x=sin^3x

sin^6x+3sin^5x+3sin^4 x+sin^3x
=(sin^2 x+sin x)^3
=(1)^3
ans=1
7.
sin x+sin^2x=1
sin x=1- sin^2 x
sin x=cos^2 x
cos^8 x+2cos^6 x+cos^4 x
=sin^4 x+2 sin^3 x+sin^2 x
=(sin^2 x+sin x)^2
=1
ans=1
10.
circumstances =2*π 15
=30π
30π=ANGLE/360°*2π*120
angle=45°

Answer 2

4) sinx = cos²x

cos²x (1 + cos²x)

= cos²x + cos⁴x = cos²x + (cos²x)²

= cos²x + sin²x = 1 { from cos²x = sinx

=========================
5)sinx + cosecx = 2
sinx + 1/sinx = 2
sin²x -2sinx +1 = 0
sinx = 1
so, cosecx = 1
hence,
sinⁿx + cosecⁿx = (1)ⁿ + (1)ⁿ = 1 + 1 = 2

=====================
6) sinx + sin²x = 1
sinx = 1 -sin²x = cos²x

now,
cos¹²x + 3cos^10x + 3cos^8x + cos^6x

={cos^4x}^3 + 3cos^8x .cos²x + 3cos⁴x .(cos²x)² + (cos²x )³

= (sin²x )³ +3(sin²x)² .cos²x + 3sin²x.(cos²x)² +(cos²x)³

=( sin²x + cos²x )³= 1 { by using sinx = cos²x }

========================
7) sinx + sin²x = 1
sinx = cos²x

cos⁴x( cos⁴x + 2cos²x +1)
=cos⁴x( cos²x + 1)²
={cos⁴x + cos²x }²
={sin²x + cos²x}² = 1

===========================
8)
sin∅1 + sin∅2 + sin∅3 = 3

we know , max value of sinx = 1
so, above possible only when
sin∅1 = 1
sin∅2 = 1
sin∅3 = 1
hence , ∅ 1= ∅2 = ∅3 = π/2

so, cos∅1 + cos∅2 + cos∅3 = 0 + 0 + 0 = 0

===========================
9)
sin10° + sin20° + sin30° + sin40°......sin60°
= { sin10°+ sin20°.....+sin180°}+ {sin190°+sin200°+ ...sin360°}

={sin10°+sin20°+..sin180°}+ { sin(180+10)+sin(180+20)+....sin(180°+180°)

=(sin10°+sin20°+sin30°....+sin180°)-(sin10°+sin20°+sin30°....sin180°}

=0

==========================
10) circumference ( length ) of 1st circle = 2π15= 30π cm

we know ,
∅ = L/r
= 30π/120 = π/4 or 45°
=========≠====================
11)
2(sin^6∅+cos^6∅)-3(sin⁴∅+cos⁴∅)+1

= 2{ 1 - 3sin²∅.cos²∅}-3{1-2sin²∅.cos²∅}+1

=0

=============================
12) cos1° + cos2° + cos3°......cos180°

=( cos1° + cos2°+ .....cos90°) + ( cos91°+ cos92°.........cos180°)

=cos1° +cos2°+....cos89°+ 0) +( -cos89°-cos88°-............-cos0°)

= -1
=====°=======°====°================

13) cot( a + b) = 0
cos(a + b) = 0 => a + b = π/2,
sin( a + b) = 1

sin( a + 2b) = sin( a + b + b)
= sin( π/2 + b) = cosb
I think some data missing , actually I can't reach final answer you can see above

14) tanA + cotA = 4

tan⁴A + cot⁴A = (tan²A + cot²A)² -2tan²A.cot²A

= { (tanA + cotA)² -2tanA.cotA}² -2
={ ( 4)² -2}² -2
=196 - 2= 194