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The correct option is (a) nπ

Easy EXPLANATION: We KNOW, TAN X = sin x / cos x. So, tan x will be zero wherever sin x is zero except the points where cos x is also zero. We know there is no point where sin x as well as cos x both are zero. So, tan x = 0 => x=nπ.

Right OPTION is (b) -5/12

To elaborate: SEC x = 13/5.

We KNOW, sec^2x – tan^2x=1

tan^2x = (13/5)^2-1 = (169/25) – 1 = 144/25

tan x = ±12/5

tan x is negative in 4^th quadrant so, tan x=-12/5

cot x = 1/tan x = 1/(-12/5) = – 5/12.

Right answer is (B) (\(\SQRT{3}\) + 1)/2\(\sqrt{2}\)

Easiest explanation: cos(15°) = cos (45°-30°) = cos45° cos30° + sin45° sin30°

= (1/\(\sqrt{2}\) * \(\sqrt{3}\)/2) + (1/\(\sqrt{2}\) * 1/2) {cos(x – y)=cos x cos y + sin x sin y}

= (\(\sqrt{3}\) +1)/2\(\sqrt{2}\).

The CORRECT OPTION is (a) π/6, 5π/6

Easiest EXPLANATION: Since cosec x is POSITIVE in 1^st and 2^nd quadrant, so there are two principal solutions to above equation i.e. π/6, π – π/6. So, x=π/6, 5π/6.

Right ANSWER is (B) False

For explanation I would say: The expression involving integer ‘N’ which GIVES all SOLUTIONS of a trigonometric equation is called the general solution. n is used in general solution only.

Correct answer is (b) 0

Easy explanation: We KNOW, COS 2x = cos^2x – sin^2x = 2cos^2x – 1 {sin^2x = 1-cos^2x}

= 2(1/\(\sqrt{2}\))^2-1 = 2(1/2) – 1 = 1-1 = 0.

The CORRECT CHOICE is (a) sec^2x

The best explanation: We KNOW, sec^2x – tan^2x=1

So, 1+ tan^2x=sec^2x.

Right OPTION is (b) √(1 – cos^2θ)/cosθ

For EXPLANATION I would say: If AOB is a righted ANGLED TRIANGLE with ∠AOB = θ and ∠BAO = 90°

Also, consider OA = x and OB = 1

By definition, cosθ = OA/OB= x/1 = x

So, AB = √(1 – x^2)

By definition, AB/OA = √(1 – x^2)/x

= √(1 – cos^2θ)/cosθ.

Correct choice is (d) 5π/4, 7π/4

For EXPLANATION: Since sin X is NEGATIVE in 3^rd and 4^th quadrant so, there are two PRINCIPAL solutions to above EQUATION i.e. x = 2π – π/4, 2π + π/4. So, x=5π/4, 7π/4.

Correct OPTION is (C) \(\SQRT{3}\)/2

For explanation I WOULD say: We know, sin 2x = (2 tan x) / (1 + tan^2x)

sin 2x=(2*1/\(\sqrt{3}\)) / (1+1/3) = (2*3)/4*\(\sqrt{3}\) = \(\sqrt{3}\)/2.

Correct ANSWER is (a) 1/2

The EXPLANATION: We know, sin^245° + cos^245°=1

So, 1- sin^245° = cos^245° = (1/√2)^2 = 1/2.

Right choice is (B) (2n+1) π/2

To EXPLAIN: When KNOW, cos X =0 whenever x is π/2, 3π/2, 5π/2, ………… i.e. all odd INTEGRAL multiples of π/2

so, x=(2n+1) π/2 when cos x=0.

Correct OPTION is (b) -1

Easiest EXPLANATION: We know, tan 3x = (3tanx – tan^3x)/(1-3tan^2x)

= (3*1 – 1)/(1-3)

= (2)/(-2) = -1.

Right choice is (a) π/3, 4π/3

For EXPLANATION I would say: SINCE cot x is POSITIVE in 1^st and 3^rd quadrant, so there are TWO principal solutions to above equation i.e. π/3, π + π/3. So, x=π/3, 4π/3.

Correct ANSWER is (B) 2-\(\sqrt{3}\)

For EXPLANATION: We know, COT (X +y) = (cot x cot y -1)/(cot y + cot x)

cot(45°+30°) = (cot 45° cot 30°-1)/(cot 45° + cot 30°)

cot 75° = (\(\sqrt{3}\) – 1)/(\(\sqrt{3}\) + 1) = 2-\(\sqrt{3}\).

The CORRECT answer is (b) 2 – \(\sqrt{3}\)

The best explanation: We KNOW, TAN (x -y) = (tan x – tan y)/(1+ tan x tan y)

tan (45°-30°) = (tan 45° – tan 30°)/(1+ tan 45° tan 30°)

tan 75° = (1- 1/\(\sqrt{3}\))/ (1+ 1/\(\sqrt{3}\)) = (\(\sqrt{3}\) – 1)/ (\(\sqrt{3}\) + 1) = 2-\(\sqrt{3}\).

The CORRECT ANSWER is (a) True

Explanation: sin (90°+x) = sin 90° COS x + cos 90° sin x {sin(x + y)=sin x cos y + cos x sin y}

= 1*cos x + 0*sin x

= cos x.

The CORRECT ANSWER is (a) 1/2

Explanation: COS(-1500°) = cos(1500°) {cos(-X) = cos x}

= cos (4*360° + 60°) = cos 60° = 1/2. {cos (2nπ+x) = cos x}

Correct OPTION is (a) ≥ 0

For EXPLANATION I would say: The GIVEN expression in LHS is,

sin^2x + cosec^2x + 2 + cos^2x + sec^2x + 2

4 + (sin^2x + cos^2x) +(1 + tan^2x) + (1 + cot^2x)

= 7 + (tan^2x + cot^2x)

= 7 + (TANX – cotx)^2 + 2 which is ≥ 0.

The CORRECT CHOICE is (c) 229°10’59”

Explanation: We know, Angle=Arc length/Radius

Angle = 40/10 = 4 radians

π radians = 180 DEGREES

1 radian = 180/π degrees

4 radians = 720/π degrees = 229.183° = 229° (0.183*60)’ = 229°(10.98)’= 229°10’59”.

The CORRECT ANSWER is (d) nπ

The explanation is: We know, cot x is not defined when sin x = 0.

sin x = 0 whenever x is 0, π, 2π, 3π, …. i.e. all integral MULTIPLES of π

so, x=nπ.

Right choice is (C) (2n+1) π/2

Easy explanation: We KNOW, tan X is not defined when cos x = 0.

cos x = 0 whenever x is π/2, 3π/2, 5π/2, ………… i.e. all odd INTEGRAL multiples of π/2

so, x=(2n+1) π/2.

The correct ANSWER is (b) tan (B/2) = (c – a)/(c + a) (cot(C – A)/2)

EASIEST explanation: According to the law of sines, in any triangle ABC,

a/SINA = b/SINB = c/sinC

So, a/b = sinA/sinB

(a + b)/(a – b) = (sinA + sinB)/( sinA – sinB)

=> (a + b)/(a – b) = (2 sin((a + b)/2) cos((a – b)/2))/ (2 sin((a + b)/2) sin((a – b)/2))

=> (a + b)/(a – b) = (tan(A + B)/2)/(tan(A – B)/2)

=> (tan(A + B)/2) = (a + b)/(a – b) (tan(A – B)/2)

=> (tan(π/2 + C/2)) = (a + b)/(a – b) (tan(A – B)/2)

=> cot (C/2) = (a + b)/(a – b) (tan(A – B)/2)

=> tan (C/2) = (a – b)/(a + b) (cot(A – B)/2)

Similarly, tan (B/2) = (c – a)/(c + a) (cot(C – A)/2).

The CORRECT choice is (c) 0 ≤ x < 2π

To explain I would say: The solutions of a TRIGONOMETRIC equation for which 0 ≤ x < 2π are CALLED principal solutions. x should lie between 0 and 2π except 2π.

The correct CHOICE is (B) -1

To explain I would SAY: We KNOW, COS 3x=4cos^3x – 3cos x

= 4(1/2)^3–3(1/2)

= 4/8 – 3/2 = 1/2 – 3/2 = -1.

Correct answer is (d) 360°

Best EXPLANATION: The angle is FORMED if we START to rotate from initial side till terminal side comes. If we start to rotate and after completing one revolution again initial side OVERLAP with terminal side, then the angle formed is 360°.

The correct answer is (a) \(\sqrt{3}\)/2

For explanation: We KNOW, sin2x = 2 sin X cos x

sin 2x = 2*1/2*\(\sqrt{3}\)/2

sin 2x = \(\sqrt{3}\)/2.

Right OPTION is (b) 12.56 cm

For explanation I would say: Radius of circle=2 cm

In 60 seconds, angle COVERED by SECOND HAND is 360°

In 40 seconds, angle covered by second hand is 360°*4/6 = 240°

240°=240*

Angle=Arc length/Radius

240*π/180 = Arc length/2

Arc length=8 π/3 = 12.56 cm.

Right choice is (b) TAN (C/2) = (a – b)/(a + b) (COT(A – B)/2)

The explanation: According to the law of sines, in any triangle ABC,

a/SINA = b/sinB = c/sinC

So, a/b = sinA/sinB

(a + b)/(a – b) = (sinA + sinB)/( sinA – sinB)

=> (a + b)/(a – b) = (2 SIN((a + b)/2) COS((a – b)/2))/ (2 sin((a + b)/2) sin((a – b)/2))

=> (a + b)/(a – b) = (tan(A + B)/2)/(tan(A – B)/2)

=> (tan(A + B)/2) = (a + b)/(a – b) (tan(A – B)/2)

=> (tan(π/2 + C/2)) = (a + b)/(a – b) (tan(A – B)/2)

=> cot (C/2) = (a + b)/(a – b) (tan(A – B)/2)

=> tan (C/2) = (a – b)/(a + b) (cot(A – B)/2).

Right choice is (a) \(\FRAC{\SQRT{3}}{\sqrt{2}}\)

For explanation I would SAY: We KNOW, sin x + sin y = 2 sin (x+y)/2 cos (x-y)/2

sin 75° + sin 15° = 2 sin (75°+15°)/2cos(75°-15°)/2

= 2 sin 45° cos 30°

= 2*1/\(\sqrt{2}\)*\(\sqrt{3}\)/2 = \(\sqrt{3}\)/\(\sqrt{2}\).

Right choice is (d) -5/3

The EXPLANATION is: SIN X=-4/5

We know, sin^2x + COS^2x=1

cos^2x = 1-(-4/5)^2 = 1-16/25=9/25

cos x=±3/5

cos x is negative in 3^rd QUADRANT so, cos x=-3/5.

sec x = 1/cos x = 1/(-3/5) = – 5/3.

The correct OPTION is (a) 1

Easy EXPLANATION: SIN(15π/6) = sin (2π + 3π/6) = sin (3π/6) {sin(2nπ+X)=sin x}

= sin (π/2) = 1.

The CORRECT choice is (a) -2

The EXPLANATION: We KNOW, cosec(-x) = cosec x

So, cosec(-30°) = -cosec 30°=-2.

The correct ANSWER is (C) 229°10’59”

The best EXPLANATION: We know, π radians = 180 DEGREES

1 radian = 180/π degrees

4 radians = 720/π degrees = 229.183° = 229° (0.183*60)’ = 229 (10.98)’ = 229°10’59”.

Right option is (B) x = nπ + π/6

Best EXPLANATION: COT x = √3

tan x = 1/√3

tan x = tan π/6

x = nπ + π/6.

The CORRECT option is (a) 0°

Easiest explanation: The angle is formed if we START to rotate from INITIAL side till terminal side comes. If they both OVERLAP then angle is said to be 0°.

Right ANSWER is (a) x = nπ/2 + (π/4)

The best explanation: tan x = COT x

tan x = cot (π/2 – x)

x = nπ + (π/2 – x)

2X = nπ + (π/2)

x = nπ/2 + (π/4).

The CORRECT ANSWER is (d) 1/√3

To EXPLAIN I WOULD SAY: tan (19π/6) = tan (2π + 7π/6) = tan 7π/6 {tan( 2nπ+x)=tan x}

= tan (π+π/6) = tan π/6 = 1/√3. {tan π+x = tan x}

Right option is (C) √2

The best explanation: We KNOW, sec(-x) = sec x

So, sec(-45°)=sec 45°=1/(cos 45°)=√2.

The CORRECT choice is (B) \(\sqrt{3}\)

For EXPLANATION: We know, sin 2x = (2 tan x) / (1 – tan^2x)

tan 2x = (2*1/\(\sqrt{3}\)) / (1-1/3) = (2*3)/2*\(\sqrt{3}\) = \(\sqrt{3}\).