Question

Points 20!

Question from ncert 10th Mathematics exersices.

Pls provide proof along with complete steps
:-)

Answer 1

LHS

tan∅/ 1 - cot∅ + cot∅/ 1 - tan∅

= sin∅/ cos∅ / 1 - cos∅/ sin∅ + cos∅ sin∅ / 1 - sin∅/ cos∅↓

used formula:

→ tan∅ = sin∅/ cos∅
→ cot∅ = cos∅ / sin∅

After lines :

= sin²∅ / cos∅( sin∅ - cos∅ ) + cos²∅/ sin∅ ( cos∅ - sin∅ )

= sin²∅/ cos∅( sin∅ - cos∅ ) - cos²∅/ sin∅ ( sin∅ - cos∅ )

→ Carefully see : we removed " + ". because we changed sin∅ ( cos∅ - sin∅ ) as sin∅ ( sin∅ - cos∅ )

After lines :

= sin∅. sin²∅ - cos∅. cos²∅/ cos∅. sin∅ ( sin∅ - cos∅ )

= sin³∅ - cos³∅ / cos∅. sin∅ ( sin∅ - cos∅ )

→ now we will use a³ - b³ = ( a - b )= ( a² + b² + ab ) formula !

= ( sin∅ - cos∅ ) ( sin²∅ + cos²∅ + sin∅. cos∅ )/ cos∅. sin∅ ( sin∅ - cos∅ )

= sin²∅ + cos∅ + sin∅.cos∅/ sin∅.cos∅

= 1 + sin∅ . cos∅/ sin∅.cos∅

→ we have used sin²∅ + cos²∅ = 1 formula

After lines :

1/ sin∅.cos∅ + sin∅.cos∅/ sin∅.cos∅

= 1 / sin∅. cos∅ + 1

= ( 1/ sin∅ ) ( 1/ cos∅ ) + 1

= cosec∅.sec∅ + 1

= 1 + sec∅.cosec∅

= RHS

we proved LHS = RHS

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