Question

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

Solution:​

Answer 1

To convert the given trigonometric ratios in terms of cot functions, use trigonometric formulas

We know that,

cosec2A – cot2A = 1

cosec2A = 1 + cot2A

Since cosec function is the inverse of sin function, it is written as

1/sin2A = 1 + cot2A

Now, rearrange the terms, it becomes

sin2A = 1/(1+cot2A)

Now, take square roots on both sides, we get

sin A = ±1/(√(1+cot2A)

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

sin2A = 1/ (1+cot2A)

Now, represent the sin function as cos function

1 – cos2A = 1/ (1+cot2A)

Rearrange the terms,

cos2A = 1 – 1/(1+cot2A)

⇒cos2A = (1-1+cot2A)/(1+cot2A)

Since sec function is the inverse of cos function,

⇒ 1/sec2A = cot2A/(1+cot2A)

Take the reciprocal and square roots on both sides, we get

⇒ sec A = ±√ (1+cot2A)/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

Answer 2

Answer:

To convert the given trigonometric ratios in terms of cot functions, use trigonometric formulas

We know that,

cosec2A – cot2A = 1

cosec2A = 1 + cot2A

Since cosec function is the inverse of sin function, it is written as

1/sin2A = 1 + cot2A

Now, rearrange the terms, it becomes

sin2A = 1/(1+cot2A)

Now, take square roots on both sides, we get

sin A = ±1/(√(1+cot2A)

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

sin2A = 1/ (1+cot2A)

Now, represent the sin function as cos function

1 – cos2A = 1/ (1+cot2A)

Rearrange the terms,

cos2A = 1 – 1/(1+cot2A)

⇒cos2A = (1-1+cot2A)/(1+cot2A)

Since sec function is the inverse of cos function,

⇒ 1/sec2A = cot2A/(1+cot2A)

Take the reciprocal and square roots on both sides, we get

⇒ sec A = ±√ (1+cot2A)/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

hope this helps you