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51.
If $sin space straight theta space plus space cos space straight theta space equals space square root of 2 space cos space straight theta$ , then the value of $left parenthesis cos space straight theta space minus space sin space straight theta right parenthesis$ is

$square root of 3 space cos space straight theta$
$square root of 3 space sin space straight theta$
$square root of 2 space cos space straight theta$
$square root of 2 space cos space straight theta$

square root of 2 space cos space straight theta

sin space straight theta space plus space cos space straight theta space equals space square root of 2 space cos space straight theta space space space space... left parenthesis straight i right parenthesis Let space cos space straight theta space minus space sin space straight theta space equals space straight x space space space space space space space space space space space space space... left parenthesis ii right parenthesis

On squaring and adding both the equations,

rightwards double arrow space space left parenthesis sin space straight theta space plus space cos space straight theta right parenthesis squared space plus space left parenthesis cos space straight theta space minus space sin space straight theta right parenthesis squared space equals space left parenthesis square root of 2 space cos space straight theta right parenthesis squared space plus space left parenthesis straight x right parenthesis squared rightwards double arrow space space sin squared straight theta space plus space cos squared straight theta space plus space 2 space sin space straight theta. space cos space straight theta space plus space cos squared straight theta space plus space sin squared straight theta space minus space 2 space sinθ. space cosθ space equals space 2 space cos squared straight theta space plus space straight x squared rightwards double arrow space space space 2 space equals space 2 space cos squared straight theta space plus space straight x squared rightwards double arrow space space space space straight x squared space equals space 2 left parenthesis 1 minus cos squared straight theta right parenthesis space equals space 2 space sin squared straight theta rightwards double arrow space space space space straight x space equals space square root of 2 space sin space straight theta space

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52.

If $straight x space sin space 45 degree space equals space straight y space cosec space 30 degree comma$ then $straight x to the power of 4 over straight y to the power of 4$ is equal to

181 Views

53.

fraction numerator sinA over denominator 1 plus cosA end fraction space plus space fraction numerator sin begin display style straight A end style over denominator 1 minus cosA end fraction

left parenthesis 0 degree less than space straight A space less than space 90 degree right parenthesis

2 cosec A
2 sec A
2 sin A
2 sin A

116 Views

54.

If $straight r space sin space straight theta space equals space 1 comma space space straight r space cos space straight theta space equals space square root of 3 comma$ then the value of $open parentheses square root of 3 space tan space straight theta space plus space 1 close parentheses$ is

$square root of 3$
$fraction numerator 1 over denominator square root of 3 end fraction$
1
1

141 Views

55.

If $sin squared straight alpha space plus space sin squared straight beta space equals 2 comma$ then the value of $cos open parentheses fraction numerator straight alpha space plus space straight beta over denominator 2 end fraction close parentheses$

1
-1
0
0

172 Views

56.

The value of
$cot straight pi over 20 space cot fraction numerator 3 straight pi over denominator 20 end fraction space cot space fraction numerator 5 straight pi over denominator 20 end fraction space cot fraction numerator 7 straight pi over denominator 20 end fraction space cot fraction numerator 9 straight pi over denominator 20 end fraction$ is

-1
$1 half$
0
0

191 Views

57.

If $sin space straight theta space plus space cos space straight theta space equals space 17 over 13 comma space space 0 less than straight theta less than 90 degree comma$ then the value of $sin space straight theta space minus space cos space straight theta$ is

$5 over 17$
$3 over 19$
$7 over 10$
$7 over 10$

151 Views

58.

If $tan space straight theta space. space tan space 2 straight theta space equals space 1 comma$ then the value of $sin squared 2 straight theta space plus space tan squared 2 straight theta$ is equal to

$3 over 4$
$10 over 3$
$3 3 over 4$
$3 3 over 4$

174 Views

59.

The expression $fraction numerator tan space 57 degree space plus space cot space 37 degree over denominator tan space 33 degree space plus space cot space 53 degree end fraction$ is equal to