If 00<x<450, then consider the following statements :Assertion (A):1+sin2x1−1+sec2x1=1+cos2x1−1+cosec2x1Reason (R) : sinx=cosxof these statements :

Consider the first expression1+sin2x1−1+sec2x1=1+sin2x1−1+cos2xcos2x=(1+sin2x)(1+cos2x)1+cos2x−cos2x−sin2xcos2x=1+(sin2xcos2x)+sin2xcos2x1+cos2x−cos2x−sin2xcos2x=2+sin2xcos2x1−sin2xcos2x ...(i)Let us now consider second expression1+cos2x1−1+sin2xsin2x=(1+sin2x)(1+cos2x)1+sin2x−sin2x−cos2xsin2x=1+(sin2xcos2x)+sin2xcos2x1−cos2xsin2x=2+sin2xcos2x1−cos2xsin2x=(ii)However sinx=cosxatx=450sinx is not equal to cosx for 0<x<450Hence reason is true assertion is wrong

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If 0^0 < x < 45^0, then consider the following statements : A

Question

If $0^{0} < x < 4 5^{0},$ then consider the following statements :

Assertion (A):

$\frac{1}{1 + s i n ^{2} x} - \frac{1}{1 + se c ^{2} x} \neq = \frac{1}{1 + co s ^{2} x} - \frac{1}{1 + cose c ^{2} x}$

Reason (R) : $s in x \neq = cos x$

of these statements :

Both A and R are true and R is the correct explanation of A

Both A and R are true, but R is not the correct explanation of A.

A is true, but R is false.

A is false, but R is true.

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0^{0} < x < 4 5^{0},

then consider the following statements :

Assertion (A):

\frac{1}{1 + s i n ^{2} x} - \frac{1}{1 + se c ^{2} x} \neq = \frac{1}{1 + co s ^{2} x} - \frac{1}{1 + cose c ^{2} x}

Reason (R) :

s in x \neq = cos x

of these statements :

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Text solutionVerified

Consider the first expression

\frac{1}{1 + s i n ^{2} x} - \frac{1}{1 + se c ^{2} x}

= \frac{1}{1 + s i n ^{2} x} - \frac{co s ^{2} x}{1 + co s ^{2} x}

= \frac{1 + co s ^{2} x - co s ^{2} x - s i n ^{2} x co s ^{2} x}{( 1 + s i n ^{2} x ) ( 1 + co s ^{2} x )}

= \frac{1 + co s ^{2} x - co s ^{2} x - s i n ^{2} x co s ^{2} x}{1 + ( s i n ^{2} x co s ^{2} x ) + s i n ^{2} x co s ^{2} x}

= \frac{1 - s i n ^{2} x co s ^{2} x}{2 + s i n ^{2} x co s ^{2} x}

...(i)
Let us now consider second expression

\frac{1}{1 + co s ^{2} x} - \frac{s i n ^{2} x}{1 + s i n ^{2} x}

= \frac{1 + s i n ^{2} x - s i n ^{2} x - co s ^{2} x s i n ^{2} x}{( 1 + s i n ^{2} x ) ( 1 + co s ^{2} x )}

= \frac{1 - co s ^{2} x s i n ^{2} x}{1 + ( s i n ^{2} x co s ^{2} x ) + s i n ^{2} x co s ^{2} x}

= \frac{1 - co s ^{2} x s i n ^{2} x}{2 + s i n ^{2} x co s ^{2} x}

= (ii)

However

s in x = cos x a t x = 4 5^{0}

s in x

is not equal to

cos x

for

0 < x < 4 5^{0}

Hence reason is true assertion is wrong

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Question Text	If $0^{0} < x < 4 5^{0},$ then consider the following statements : Assertion (A): $\frac{1}{1 + s i n ^{2} x} - \frac{1}{1 + se c ^{2} x} \neq = \frac{1}{1 + co s ^{2} x} - \frac{1}{1 + cose c ^{2} x}$ Reason (R) : $s in x \neq = cos x$ of these statements :
Answer Type	Text solution:1
Upvotes	150

If 00<x<450, then consider the following statements :Assertion (A):1+sin2x1​−1+sec2x1​=1+cos2x1​−1+cosec2x1​Reason (R) : sinx=cosxof these statements :

Text solutionVerified

Practice questions from similar books

If $0^{0} < x < 4 5^{0},$ then consider the following statements :

Assertion (A):

$\frac{1}{1 + s i n ^{2} x} - \frac{1}{1 + se c ^{2} x} \neq = \frac{1}{1 + co s ^{2} x} - \frac{1}{1 + cose c ^{2} x}$

Reason (R) : $s in x \neq = cos x$

of these statements :