Prove that cosec 60 degrees cot 30 degrees tan 60 degrees=2secsquared 45degrees cos 30 degree
Work on the left-hand side first:
csc60 cot30 tan60
=(1/sin60)(tan60)(tan60)
=(1/sin60)(tan²60)
=(1/sin60)(sec²60-1)
=(2/√3)(2²-1)
=2√3
Work on the right-hand side:
2sec²45cos30
=2(2/√2)² (√3)/2
=8/2 (√3)/2
=2√3
Thus LHS=RHS and the proof is complete.
csc60° * cot 30° * tan 60°
= 2/√3 * √3 * √3
= 2√3
2sec^2 45° cos30°
= 2*2*√3/2
= 2√3
To prove cosec 60 degrees cot 30 degrees tan 60 degrees = 2 sec² 45 degrees cos 30 degrees, we will simplify each side of the equation step by step.
Left-hand side (LHS):
cosec 60 degrees = 1/sin 60 degrees
cot 30 degrees = 1/tan 30 degrees
tan 60 degrees = sin 60 degrees / cos 60 degrees
cosec 60 degrees cot 30 degrees tan 60 degrees = (1/sin 60 degrees) * (1/tan 30 degrees) * (sin 60 degrees / cos 60 degrees)
= (cos 30 degrees / sin 30 degrees) * (1/sin 30 degrees) * (sin 60 degrees / cos 60 degrees)
= cos 30 degrees * sin 60 degrees
= (sqrt(3)/2) * (sqrt(3)/2)
= 3/4
Right-hand side (RHS):
2 sec² 45 degrees cos 30 degrees = 2 * [(1/cos 45 degrees)²] * cos 30 degrees
= 2 * (sqrt(2)/2)² * (sqrt(3)/2)
= 2 * (1/2) * (sqrt(3)/2)
= sqrt(3)/2
Since LHS = 3/4 and RHS = sqrt(3)/2, the equation is not true. Therefore, the statement is false and the given expression is not equal to 2 sec² 45 degrees cos 30 degrees.
To prove this trigonometric identity, we'll start by simplifying each side of the equation separately using the fundamental trigonometric identities.
Left-hand side (LHS):
cosec(60°) * cot(30°) * tan(60°)
Recall the trigonometric identities:
cosec(θ) = 1/sin(θ)
cot(θ) = 1/tan(θ)
Using these identities, we can rewrite the left-hand side as:
(1/sin(60°)) * (1/tan(30°)) * tan(60°)
1/(sin(60°) * tan(30°)) * tan(60°)
Now, we need to simplify sin(60°) and tan(30°):
sin(60°) = √3/2
tan(30°) = 1/√3
Substituting these values back into the equation:
1/((√3/2) * (1/√3)) * tan(60°)
1/(√3/2) * tan(60°)
Now, tan(60°) = √3, so we can substitute this value:
1/(√3/2) * √3
2/√3 * √3
2
Therefore, the left-hand side (LHS) simplifies to 2.
Right-hand side (RHS):
2sec^2(45°) * cos(30°)
Recall the trigonometric identity:
sec(θ) = 1/cos(θ)
Using this identity, we can rewrite the right-hand side as:
2(1/cos(45°))^2 * cos(30°)
2(1/(√2))^2 * cos(30°)
2(1/2) * cos(30°)
Simplifying further:
2/2 * cos(30°)
1 * cos(30°)
cos(30°)
Now, cos(30°) = √3/2, so we can substitute this value:
√3/2
Therefore, the right-hand side (RHS) simplifies to √3/2.
Since the left-hand side (LHS) simplifies to 2 and the right-hand side (RHS) simplifies to √3/2, we can conclude that:
2 = √3/2
Unfortunately, these two expressions are not equal, which means that the initial trigonometric identity you provided is not true.