prove that: (sin tita -cosec tita)² + (cos tita)² =cot²+tan²tita - 1?
Answers
Step-by-step explanation:
(You have missed that -secθ in the question. pls check the question. It should be prove that (sinθ -cosecθ)² + (cosθ-secθ)² = cot²θ+tan²θ - 1 )
Proof:
To prove: (sinθ -cosecθ)² + (cosθ-secθ)² = cot²θ+tan²θ - 1
take LHS
= (sinθ -cosecθ)² + (cosθ-secθ)²
= sin²θ - 2sinθcosecθ + cosec²θ + cos²θ - 2cosθsecθ + sec²θ
= sin²θ+cos²θ -2(1) + cosec²θ -2 + sec²θ
= 1-4+cosec²θ + sec²θ
= cosec²θ-1 + sec²θ - 1 -1
= cot²θ + tan²θ - 1
= cot²θ - sec²θ ----------- (1)
take RHS
= cot²θ+tan²θ - 1
= cot²θ - (1-tan²θ)
= cot²θ - sec²θ --------------(2)
from (1) & (2),
=> LHS = RHS
Hence proved
= (sinθ -cosecθ)² + (cosθ-secθ)²
= sin²θ - 2sinθcosecθ + cosec²θ + cos²θ - 2cosθsecθ + sec²θ
= sin²θ+cos²θ -2(1) + cosec²θ -2 + sec²θ
= 1-4+cosec²θ + sec²θ
= cosec²θ-1 + sec²θ - 1 -1
= cot²θ + tan²θ - 1
= cot²θ - sec²θ