Write all the other trigonometric ratios of ∠A in terms of sec A
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Answer:
Write all the other trigonometric ratios of ∠A in terms of sec A.
Step-by-step explanation:
An equation involving trigonometric ratios of an angle is called a trigonometric identity. This is true for all angles A such that 0°≤A≤90°.
Some identities are:
1) cos² A + sin²A = 1
2) sec² A - tan² A = 1.
3) cosec² A - cot² A = 1.
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We know that,
sec A = 1/cos A
⇒ cos A = 1/sec A
cos²A + sin²A = 1
⇒ sin²A = 1 – cos²A
⇒ sin²A = 1 – (1/sec²A) (cosA= 1/secA)
⇒ sin²A = (sec²A-1)/sec²A
⇒sinA = √((sec²A-1)/sec²A)
⇒sinA = √(sec²A-1) ÷ (secA)...(i)
sin A = 1/cosec A
⇒ cosec A = 1/sin A
⇒cosecA= secA ÷√sec²A-1 (from eq i)
Now,
sec²A – tan²A = 1
⇒ tan²A = sec²A + 1
⇒tanA = √sec²A + 1...(ii)
tan A = 1/cot A
⇒ cot A = 1/tan A
⇒cotA = 1/√sec²A + 1 (from eq ii)
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