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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