Cos square 30 degree cos square 45 degree + 4 sec square 60 degree + 1/2 cos square 90 degree - 2 tan square 60 degree
cos^2 30° cos^2 45° + 4sec^2 60° + (1/2)cos^2 90° - 2tan^2 60°
these are all special angles and you should memorize the trig ratios of the 30-60-90 and the 45-45-90 triangles
= (√3/2)^2 (1/√2)^2 + 4(2)^2 + (1/2)(0) - 2(√3)^2
= (3/4)(1/2) + 16 + 0 - 2(3)
= 3/8 + 16 - 6
= 10 3/8 or 83/8
To compute the value of the expression, we will break it down step by step:
1. Start with the expression:
cos^2(30 degrees) * cos^2(45 degrees) + 4sec^2(60 degrees) + 1/2 * cos^2(90 degrees) - 2tan^2(60 degrees)
2. Simplify each term individually:
a. cos^2(30 degrees) = (sqrt(3)/2)^2 = 3/4
Here, we use the trigonometric identity cos^2(x) = (cos(x))^2 and substitute the value of cos(30 degrees) as sqrt(3)/2.
b. cos^2(45 degrees) = (1/sqrt(2))^2 = 1/2
Again, we use the trigonometric identity cos^2(x) = (cos(x))^2 and substitute the value of cos(45 degrees) as 1/sqrt(2).
c. sec^2(60 degrees) = (1/cos(60 degrees))^2 = (2/3)^2 = 4/9
Here, we use the trigonometric identity sec^2(x) = (1/cos(x))^2 and substitute the value of cos(60 degrees) as 1/2.
d. cos^2(90 degrees) = 0
Since cos(90 degrees) = 0, the square of this value will also be 0.
e. tan^2(60 degrees) = (sin(60 degrees)/cos(60 degrees))^2 = (sqrt(3)/2)^2 = 3/4
We use the trigonometric identity tan^2(x) = (sin(x)/cos(x))^2 and substitute the values of sin(60 degrees) and cos(60 degrees).
3. Substitute the simplified values back into the expression:
(3/4)(1/2) + 4(4/9) + 1/2(0) - 2(3/4)
4. Further simplify the expression:
3/8 + 16/9 + 0 - 3/2
5. Find the common denominator for all the fractions:
LCM(8, 9, 2) = 72
(27/72) + (128/72) + 0 - (108/72)
(27 + 128 - 108)/72
47/72
Therefore, the value of the given expression is 47/72.
Thanks for help ....... Good .... Nice experience 🥰🥰
anonymous cos 90 is 0 itself
Reiny gave the correct answer
the answer is right