Question
Question: If \( \sin \theta -\cos \theta =0 \), then \( {{\sin }^{4}}\theta +{{\cos }^{4}}\theta = \)? A. 1 ...
If sinθ−cosθ=0, then sin4θ+cos4θ=?
A. 1
B. 21
C. 41
D. 43
Solution
It will be very complicated to expand (a+b)4.
Find the value of θ which satisfies the given equation and then substitute it in the given expression to evaluate it.
Recall that sin45∘=21 and cos45∘=21.
Complete step by step answer:
We are given the equation:
sinθ−cosθ=0
⇒ sinθ=cosθ
Dividing both sides by cosθ, we get:
⇒ cosθsinθ=1
We know that cosθsinθ=tanθ, therefore, we get:
⇒ tanθ=1
We also know that tan45∘=1, therefore, θ=45∘.
Now, substituting sin45∘=21 and cos45∘=21 in the given expression, we get:
sin4θ+cos4θ
= (21)4+(21)4
= 41+41
= 42
= 21
The correct answer is B. 21.
Note: The value of sinθ and cosθ lies between -1 and 1.
sin(−θ)=−sinθ and cos(−θ)=cosθ.
Values of Trigonometric Ratios for Common Angles:
| 0°| 30°| 45°| 60°| 90°
---|---|---|---|---|---
sin| 0| 21 | 21 | 23 | 1
cos| 1| 23 | 21 | 21 | 0
tan| 0| 31 | 1| 3 | ∞
csc| ∞ | 2| 2 | 32 | 1
sec| 1| 32 | 2 | 2| ∞
cot| ∞ | 3 | 1| 31 | 0