Trigonometric identities are equations that apply to various trigonometric functions and are true for all values of the variable in the domain. These are the equation’s equalities for all possible values of the variables. The trigonometric ratios used in these identities are sine, cosine, tangent, cosecant, secant, and cotangent. All of these trigonometric ratios are calculated using right triangle sides such as an adjacent side, opposing side, and hypotenuse side. Only a right-angle triangle can use these trigonometric identities.
Cosec Cot formula
The cosec cot formula is a Pythagorean trigonometric identity in trigonometry since it is based on Pythagoras’ theorem. It says that at every angle, the square of cosecant is equal to the sum of the square of cotangent and unity.
cosec2 θ = 1 + cot2 θ
Derivation
Â
Consider a right triangle ABC with angle θ between its base and hypotenuse.
Applying Pythagoras theorem on this triangle, we get
AC2 = AB2 + BC2 Â Â Â Â Â Â
Dividing both sides by AB2, we get
AC2/AB2 = AB2/AB2 + BC2/AB2
(AC/AB)2 = 1 + (BC/AB)2 Â Â ……. (1)
We know, for angle θ,
cosec θ = Hypotenuse/Perpendicular
cosec θ = AC/AB         ……. (2)
Also, we have
cot θ = Base/Perpendicular
cot θ = BC/AB          ……. (3)
Using (2) and (3) in (1), we get
cosec2 θ = 1 + cot2 θ
This proves the cosec cot formula.
Sample Problems
Problem 1. If cot θ = 3/4, find the value of cosec θ using the formula.
Solution:
We have,
cot θ = 3/4
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (3/4)2
cosec2 θ = 1 + 9/16
cosec2 θ = 25/16
cosec θ = 5/4
Problem 2. If cot θ = 12/5, find the value of cosec θ using the formula.
Solution:
We have,
cot θ = 12/5
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (12/5)2
cosec2 θ = 1 + 144/25
cosec2 θ = 169/25
cosec θ = 13/5
Problem 3. If cos θ = 4/5, find the value of cosec θ using the formula.
Solution:
We have, cos θ = 4/5.
Clearly sin θ = 3/5. Hence we have, cot θ = 4/3.
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (4/3)2
cosec2 θ = 1 + 16/9
cosec2 θ = 25/9
cosec θ = 5/3
Problem 4. If sin θ = 12/13, find the value of cosec θ using the formula.
Solution:
We have, sin θ = 12/13.
Clearly cos θ = 5/13. Hence we have, cot θ = 12/5.
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (12/5)2
cosec2 θ = 1 + 144/25
cosec2 θ = 169/25
cosec θ = 13/5
Problem 5. If sin θ = 4/5, find the value of cot θ using the formula.
Solution:
We have, sin θ = 4/5.
Clearly cosec θ = 5/4.
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cot2 θ = (5/4)2 – 1
cot2 θ = 25/16 – 1
cot2 θ = 9/16
cot θ = 3/4
Problem 6. If sec θ = 17/8, find the value of cosec θ using the formula.
Solution:
We have, sec θ = 17/8.
Clearly cos θ = 8/17. Hence we have, cot θ = 8/15.
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (8/15)2
cosec2 θ = 1 + 64/225
cosec2 θ = 289/225
cosec θ = 17/15
Problem 7. If tan θ = 12/5, find the value of cosec θ using the formula.
Solution:
We have, tan θ = 12/5.
Hence we have, cot θ = 5/12.
Using the formula we have,
cosec2 θ = 1 + cot2 θ
cosec2 θ = 1 + (5/12)2
cosec2 θ = 1 + 25/144
cosec2 θ = 169/144
cosec θ = 13/12
Last Updated :
10 Jan, 2024
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