Question

cot a - cos a/cot a + cos a = sec²a + tan²a - 2 sec a × tan a
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Answer 1

Step-by-step explanation:

the above picture gives the explanation

Answer 2

By expanding right and left hand side of the given equation we prove that, cot a - cos a/cot a + cos a = sec²a + tan²a - 2 sec a × tan a

We may begin by making the left side (L.H.S) of the equation simpler:

(cot a - cos a) / (cot a + cos a)

= [(cos a / sin a) - cos a] / [(cos a / sin a) + cos a] (using the definitions of cot a and cos a)

= [(cos a - cos a × sin a) / sin a] / [(cos a + cos a × sin a) / sin a]

= (cos a - cos a × sin a) / (cos a + cos a × sin a)

= cos a (1 - sin a) / cos a (1 + sin a)

= (1 - sin a) / (1 + sin a)

Let's now make the right-hand side (R.H.S) of the equation simpler:

sec²a + tan²a - 2 sec a × tan a

= (1/cos²a) + (sin²a / cos²a) - 2 (1/cos a) (sin a / cos a)

= (1 + sin²a - 2 sin a) / cos²a

= (1 - sin a)² / (1 + sin a)²

Currently, we can observe that the left and right sides are equal:

=> (1 - sin a) / (1 + sin a) = (1 - sin a)² / (1 + sin a)²

Multiplying both sides by (1 + sin a)², we get:

=> (1 - sin a) (1 + sin a) = (1 - sin a)²

By expanding the left-hand side, we get:

=> 1 - sin²a = 1 - 2sin a + sin²a

By simplifying, we get:

=> 1 - sin²a = 1 - 2sin a + sin²a

Hence, we have proved that:

cot a - cos a/cot a + cos a = sec²a + tan²a - 2 sec a × tan a

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