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Q.

If $45 ° - 45 ° = x \cos \cos 45 ° \sin 45 °$ then value of $x$ is equal to

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a

2

b

\frac{1}{2}

c

- \frac{1}{2}

d

-

1

answer is D.

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Detailed Solution

We know from the trigonometric ratios in a right angled triangle the sine of any angle is given by the ratio of length of side opposite to the angle to the length of hypotenuse. In the figure the sine of the angle θ is given by sinθ=

\frac{p}{h}

Similarly the cosine of an angle is the ratio of length of side adjacent to the angle (excluding hypotenuse) to the length of hypotenuse. So we have cosine of angle θ as
cosθ=

\frac{b}{h}

We know from the basic sine and cosine values of angles of measures

30 °

,

45 °, 60 °

as
sin

30 °

=

\frac{1}{2}

, sin

45 °

=

\frac{1}{\sqrt 2}

, sin

60 °

=

\frac{\sqrt 3}{2}

cos

30 °

=

\frac{\sqrt 3}{2}

, cos

45 °

=

\frac{1}{\sqrt 2}

, cos

60 °

=

\frac{1}{2}

We are asked to find the values of xx from the following equation,

45 ° - 45 ° = x \cos \cos 45 ° \sin 45 °

⇒

{(\cos \cos 45 °)}^{2} - {30 °)}^{2} = x \cos \cos 45 ° \sin 45 °

We put the known values of sine and cosine in the above equation to have;
⇒

{(\frac{1}{\sqrt 2})}^{2} - {(\frac{\sqrt{3}}{2})}^{2} = x (\frac{1}{\sqrt{2}}) (\frac{1}{\sqrt{2}})

⇒

\frac{1}{2} - \frac{3}{4}

=

x \frac{1}{\sqrt{2} \sqrt{2}}

⇒

\frac{1 - 3}{4}

=

x \frac{1}{2}

⇒

\frac{- 1}{2}

=

x \frac{1}{2}

⇒x=−1
So the option is 4.

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