Step-by-step explanation:
<h3><u>Given :-</u></h3>
[1+(1/Tan²θ)] + [ 1+(1/Cot²θ)]
<h3>
<u>Required To Prove :-</u></h3>
[1+(1/Tan²θ)]+[1+(1/Cot²θ)] = 1/(Sin²θ-Sin⁴θ)
<h3><u>Proof :-</u></h3>
On taking LHS
[1+(1/Tan²θ)] + [ 1+(1/Cot²θ)]
We know that
Tan θ = 1/ Cot θ
and
Cot θ = 1/Tan θ
=> (1+Cot²θ)(1+Tan²θ)
=> (Cosec² θ) (Sec²θ)
Since Cosec²θ - Cot²θ = 1 and
Sec²θ - Tan²θ = 1
=> (1/Sin² θ)(1/Cos² θ)
Since , Cosec θ = 1/Sinθ
and Sec θ = 1/Cosθ
=> 1/(Sin²θ Cos²θ)
We know that Sin²θ+Cos²θ = 1
=> 1/[(Sin²θ)(1-Sin²θ)]
=> 1/(Sin²θ-Sin²θ Sin²θ)
=> 1/(Sin²θ - Sin⁴θ)
=> RHS
=> LHS = RHS
<u>Hence, Proved.</u>
<h3><u>Answer:-</u></h3>
[1+(1/Tan²θ)]+[1+(1/Cot²θ)] = 1/(Sin²θ-Sin⁴θ)
<h3><u>Used formulae:-</u></h3>
→ Tan θ = 1/ Cot θ
→ Cot θ = 1/Tan θ
→ Cosec θ = 1/Sinθ
→ Sec θ = 1/Cosθ
<h3><u>Used Identities :-</u></h3>
→ Cosec²θ - Cot²θ = 1
→ Sec²θ - Tan²θ = 1
→ Sin²θ+Cos²θ = 1
Hope this helps!!
Step-by-step explanation:
Since, r varies directly as s
Answer:
257 is prime.
Step-by-step explanation:
To evaluate if a number is prime, we just need to evaluate it for the prime numbers that are equal or lesser than the said number's square root.
In this case, √257 = 16.03 so we just need to see if 257 is divisible by <u>2, 3, 5, 7, 11 and 13</u> (the prime numbers that come before 16)
- 257 is odd, so it is not divisible by 2.
- The sum of its digits is 14, therefore, it is not divisible by 3.
- 257 ends in 7, therefore it's not divisible by 5.
- 257/ 7 = 36.71 so it's not divisible by 7.
- 257/ 11 = 23.36 so it's not divisible by 11
- Finally 257 / 13= 19.76 so it's not divisible by 13.
Therefore, 257 is prime.
Answer:
Area of a Square it base x height so 4x4=16+16 = 32 in^2
Step-by-step explanation:
