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Question : What are Reciprocal of trigonometric functions. Use them to find exact values of the following expressions
- tan 45° − 1/ cot 45°
- cot 30° ⋅ sec 30° ⋅ sin 30°
Answer :
Required Reciprocal of trigonometric functions are the following
- sin 𝜃 = 1/(csc 𝜃)
- csc 𝜃 = 1/(sin 𝜃)
- cos 𝜃 = 1/(sec 𝜃)
- sec 𝜃 = 1/(cos 𝜃)
- tan 𝜃 = 1/(cot 𝜃)
- cot 𝜃 = 1/(tan 𝜃)
where following are known as Quotient formulas
tan 𝜃 = (sin 𝜃)/(cos 𝜃)
cot 𝜃 = (cos 𝜃)/(sin 𝜃)
To find exact values of the expressions
- tan 45° − 1/ cot 45°
Since tan 𝜃 = 1/(cot 𝜃)
Hence tan 45° = 1/( cot 45°)
therefore 1/ cot 45° − 1/ cot 45° = 0
- cot 30° ⋅ sec 30° ⋅ sin 30°
Since sec 𝜃 = 1/(cos 𝜃) , cot 𝜃 = cos 𝜃 /(sin 𝜃)
hence cot 30° ⋅ sec 30° ⋅ sin 30° = ( cos 30°/ sin 30°) (1/cos 30°) ⋅ sin 30° =1
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