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## SSC Shortcuts (Substitution method)

At times students find trigonometry questions in SSC paper very hard and they leave those questions unattempted. Whereas, there are other students who are well versed with shortcut methods. These students use shortcuts and solve such questions within seconds. Here we are discussing some shortcuts tips and tricks for our students so that next time you are in a position to solve complex questions within seconds.

Q19) If cos ⁡x + cos ⁡y = 2 then the value of sin ⁡x + sin ⁡y is

1)
0
2) 1
3) 2
4) -1

cos ⁡x + cos ⁡ y = 2
cos ⁡ x = 1; cos ⁡ y = 1
x = y = 0
sin ⁡ x + sin ⁡ y = 0
Hence, option 1.

Q20) The value of is

1)
5
2) 4
3) 3
4) 2

Easiest way to solve such questions is to substitute value of θ.
Put θ = 45°

Hence, option 1.

Q21) If and ⁡ then cos2α is

1)

2)
3)
4)

Full method to solve this question:

Shortcut:
Put α = 60°; β = 30
tan⁡60° = n tan30° i.e. n = 3 or n2 = 9
sin60° = m sin30° i.e. m = √3 or m2 = 3
cos2⁡α = cos260° = 1/4
For m2 = 3; n2 = 9
Option 1: 1/3 Eliminated.
Option 2: 1/4 Satisfied.
Option 3: 1/2 Eliminated.
Option 4: 3/10 Eliminated.
Hence, option 2.

Trigonometric Identities

In the last question we got a unique answer using substitution method. However, it may not always be true. At times using, substitution method we may get more than one option. So Lets discuss how to solve such questions.

Q22) If then the value of is

Use substitution method, put θ = 90° Answer is either option 3 or option 4.
As we did not get a unique answer so put θ=0° Answer is option 4.
Hence, option 4.

Q23) If then the value of

1)
9
2) 0
3) 1
4) 4

Easiest substitution for this question is
Put θ = 45°,Φ = 45°
x = a sec45° cos45° = a
y = b sec45° sin⁡45° = b
z = c tan45° = c

Hence, option 3.

Q24) If sin⁡ θ + cos⁡θ = x then the value of sin6⁡θ + cos6⁡θ is equal to

Use substitution method, put
Answer is either option 3 or option 4.
Now, put
Answer is either option 1 or option 3.
So, common answer from the above substitutions is option 3.
Hence, option 3.

Q25) If sinθ + cosθ = 1, what is the value of sinθ cosθ

1)
2
2) 0
3) 1
4) 1/2

sinθ + cosθ = 1
(sinθ + cosθ)2 = 1
sin2θ + cos2θ + 2sinθ cosθ = 1
1 + 2sinθ cosθ = 1 i.e. sinθ cosθ = 0

Shortcut:
sinθ + cosθ = 1 so, put θ = 0°
sinθ cosθ = sin0° cos0° = 0.
Hence, option 2.

Q26) The numerical value of is

1)
0
2) -1
3) 1
4) 2

Put θ = 45°
Hence, option 3.

Q27) If cosθ + secθ = 2, then the value of cos6θ + sec6θ is

1)
1
2) 2
3) 4
4) 8

cosθ + secθ = 2
cosθ + 1/cosθ = 2
So, cosθ = 1 i.e. θ =
0° cos6θ + sec6θ = cos60° + sec60° = 2
Hence, option 2.

Q28) If sinθcosθ = 1/2, then what is sin6θ + cos6θ equal to

1)
1
2) 2
3) 3
4) 1/4

sinθcosθ = 1/2
2sinθcosθ = 1
sin2θ = 1
2θ = 90°
θ = 45°
sin6θ + cos6θ = sin645° + cos645° = 1/4
Hence, option 4.