Question

Evaluate (cosecA - sin A) (sec A-cos A) (tan A+cot A)​

Answer 1

Answer:

L. H. S =

(cosecA - sinA)×(secA - cosA)

= (1/sinA - sinA)×(1/cosA - cosA)

= (1-sin²A)/sinA × (1-cos²A)/cosA

= cos²A/sinA × sin²A/cosA

= cos²Asin²A/cosAsinA

= cosAsinA

Now, R. H. S =

1/(tanA + cotA)

= 1/(sinA/cosA + cosA/sinA)

= 1/[(sin²A + cos²A)/cosAsinA]

= 1/[(1/cosAsinA)]

= cosAsinA

Since L. H. S. = R. H. S.

Therefore the given equation is proof.

Answer 2

Answer:

$= (coseca - sina)(seca - cosa)(tana + cota) \\ = ( \frac{1}{sina} - sina)( \frac{1}{cosa} - cosa)( \frac{sina}{cosa} + \frac{cosa}{sina} ) \\ = ( \frac{1 - sin {}^{2}a }{sina} )( \frac{1 - cos {}^{2}a }{cosa} )( \frac{sin {}^{2}a + cos {}^{2}a }{sinacosa} ) \\ = \frac{cos {}^{2}a }{sina} \times \frac{sin {}^{2}a }{cosa} \times \frac{1}{sinacosa} \\ = \frac{cos {}^{2}a \times sin {}^{2} a }{cos {}^{2}a \times sin {}^{2} a} \\ = 1 \: \: answer$