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

Asked by Abhisek | 1 year ago |  53

#### 1 Answer

##### Solution :-

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

$$\dfrac{1}{(1+sin^2A)}$$ = 1 + cot2A

Now, rearrange the terms, it becomes

sin2A = $$\dfrac{1}{(1+cot^2A)}$$

Now, take square roots on both sides, we get

sin A = $$\dfrac{±1}{\sqrt{(1+cot^2A)}}$$

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 = $$\dfrac{1}{(1+cot^2A)}$$

Now, represent the sin function as cos function

1 – cos2A =$$\dfrac{1}{(1+cot^2A)}$$

Rearrange the terms,

cos2A = 1 – $$\dfrac{1}{(1+cot^2A)}$$

⇒cos2A = $$1-\dfrac{(1+cot^2A)}{(1+cot^2A)}$$

Since sec function is the inverse of cos function,

$$\dfrac{1}{sec^2A }$$$$1-\dfrac{(cot^2A)}{(1+cot^2A)}$$

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

⇒ sec A = $$±\dfrac{\sqrt{(1+cot^2A)}}{cotA}$$

Now, to express tan function in terms of cot function

tan A = $$\dfrac{ sin A}{cos A}$$ and cot A = $$\dfrac{cos A}{sin A}$$

Since cot function is the inverse of tan function, it is rewritten as

tan A =$$\dfrac{1}{cot A}$$

Answered by Pragya Singh | 1 year ago

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