Question

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Answer 1

Final Answer :
24 :)B
25:)C ( already done)
26 :)A

Steps:
24 :
1) We know tan theta will be negative, since it lies in 4th Quadrant, so will equate

tana =- tanα ,
where a is theta angle, and α is such that given expression follows.
=> tanα > 0

2)Since, tanα is positive so we will easily do the comparison of powers.

For Calculation see pic 1
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Steps:26:)
Formulas used here:
1) 2cosAsinB= sin(A+B)-sin(A-B)
2) sin(π-A) = sin(A)

Steps:
1) Let the given summation be termed as :S:.
2) Multiple S by sin(π/19) in both sides.
3) We will use formula in(1) to convert them in difference such that two consecutive term elements will cancel each other.
4) Cancel the terms and then use formula in step 2
For Calculation see pic 2 .

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I made one nice formula for these summation for n terms in AP.
Check out in last pic. 3

Answer 2

Heya!

25)

A= cos² x + sin^4 x
=cos² x(cos² x + sin² x) + sin^4 x
=(cos^4 x + sin^4 x + 2 cos² x sin² x) - cos² x sin² x
= (cos² x + sin² x)^²- cos² x sin² x
= 1 - cos² x sin² x
= 1 - 1/4 (2 cos x sin x)²
= 1 - 1/4 sin² (2x)

Now -1 <= sin theta <= 1, so 0<= sin² theta <=1
Hence A can take values between 3/4 to 1, both inclusive.