Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Dear Student

To convert the given trigonometric ratios in terms of cot functions, use trigonometric formulas[page19image3424289280]We know that,
cosec^2 A - cot^2 A = 1
cosec^2 A = 1 + cot^2 A
Since cosec function is the inverse of sin function, it is written as

1/sin^2 A = 1 + cot^2 A

Now, rearrange the terms, it becomes

sin^2 A = 1/(1+cot^2 A)
Now, take square roots on both sides, we get

sin A = ±1/(√(1+cot^2 A)
The above equation defines the sin function in terms of cot function

Now, to express sec function in terms of cot function, use this formula

sin^2 A = 1/(1+cot^2 A)
Now, represent the sin function as cos function

1 - cos^2 A = 1/(1+cot^2 A) Rearrange the terms,

cos^2 A = 1 - 1/(1+cot^2 A)

⇒cos^2 A = (1-1+cot^2 A)/(1+cot^2 A)
Since sec function is the inverse of cos function,

⇒1/sec^2 A = cot^2 A/(1+cot^2 A)
Take the reciprocal and square roots on both sides, we get

⇒sec A =±√(1+cot^2 A)/cotA
Now, to express tan function in terms of cot function

tan A = sin A/cos A and cot A = cos A/sin A
Since cot function is the inverse of tan function, it is rewritten as

tan A = 1/cot A

Thanks