71.
The numerical value of $$\frac{1}{{1 + {{\cot }^2}\theta }}$$ ÃÂÃÂ + $$\frac{3}{{1 + {\text{ta}}{{\text{n}}^2}\theta }}$$ ÃÂÃÂ + $$2{\sin ^2}\theta $$ ÃÂÃÂ will be?
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Solution:
$$\eqalign{ & \frac{1}{{1 + {{\cot }^2}\theta }} + \frac{3}{{1 + {\text{ta}}{{\text{n}}^2}\theta }} + 2{\sin ^2}\theta \cr & \Rightarrow \frac{1}{{{{\operatorname{cosec} }^2}\theta }} + \frac{3}{{{{\sec }^2}\theta }} + 2{\sin ^2}\theta \cr & \Rightarrow {\sin ^2}\theta + 3{\text{co}}{{\text{s}}^2}\theta + 2{\sin ^2}\theta \cr & \Rightarrow 3\left( {{{\sin }^2}\theta + {\text{co}}{{\text{s}}^2}\theta } \right) \cr & \Rightarrow 3\left( 1 \right) \cr & \Rightarrow 3 \cr} $$
72.
If $${\text{cos}}\theta = \frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}$$ ÃÂÃÂ ÃÂÃÂ then the value of$${\text{cot}}\theta $$ÃÂÃÂ is equal to $$\left[ {{\text{if }}{0^ \circ } \leqslant \theta \leqslant {{90}^ \circ }} \right]$$
(A) $$\frac{{2xy}}{{{x^2} - {y^2}}}$$
(B) $$\frac{{2xy}}{{{x^2} + {y^2}}}$$
(C) $$\frac{{{x^2} + {y^2}}}{{2xy}}$$
(D) $$\frac{{{x^2} - {y^2}}}{{2xy}}$$
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Solution:
$${\text{cos}}\theta = \frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}$$ AC2 = (x2 + y2)2 - (x2 - y2)2 = x4 + y4 + 2x2y2 - x4 - y4 + 2x2y2 = 4x2y2 ⇒ AC = 2xy ⇒ cotθ = $$\frac{{{x^2} - {y^2}}}{{2xy}}$$
73.
ABCD is a rectangle of which AC is a diagonal. The value of (tan ÃÂÃÂÃÂâÃÂÃÂÃÂÃÂÃÂàCAD + 1)sin ÃÂÃÂÃÂâÃÂÃÂÃÂÃÂÃÂàBAC is?22
(A) 2
(B) $$\frac{1}{4}$$
(C) 1
(D) 0
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Solution:
= (tan2α + 1 )sin2β = (tan245° + 1 )sin245° $$ = (1 + 1) {\left( {\frac{1}{{\sqrt 2 }}} \right)^2}$$ $$ = 2 \times \frac{1}{2} = 1$$
74.
The simplified value of (sec x sec y + tan x tan y)2 - (sec x tan y + tan x sec y)2
(A) -1
(B) 0
(C) sec2 x
(D) 1
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Solution:
(sec x sec y + tan x tan y)2 - (sec x tan y + tan x sec y)2 = sec2x. sec2y + tan2x. tan2y + 2sec x. sec y. tan x. tan y - sec2x. tan2y - tan2x. sec2y + 2sec x. tan y. tan x sec y = sec2x [sec2y - tan2y] - tan2x [sec2y - tan2y] = (sec2x - tan2x) (sec2y - tan2y) = 1 × 1 = 1
75.
sin4 ÃÂÃÂÃÂø + cos4 ÃÂÃÂÃÂø in terms of sinÃÂÃÂÃÂø can be written as:
(A) 2sin4θ + 2sin2θ - 1
(B) 2sin4θ - 2sin2θ
(C) 2sin4θ - 2sin2θ - 1
(D) 2sin4θ - 2sin2θ + 1
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Solution:
(sin2θ + cos2θ)2 = 1 sin4θ + cos4θ + 2sin2θ.cos2θ = 1 sin4θ + cos4θ = 1 - 2sin2θ.cos2θ sin4θ + cos4θ = 1 - 2sin2θ(1 - sin2θ) sin4θ + cos4θ = 1 - 2sin2θ + 2sin4θ
76.
The value of $$\frac{{\sec \theta \left( {\sin \theta - 2{{\sin }^3}\theta } \right)}}{{2{{\cos }^3}\theta - \cos \theta }}$$ ÃÂÃÂ ÃÂÃÂ is:
(A) tanθ.cosθ
(B) secθ.sinθ
(C) secθ.tanθ
(D) sinθ.cosθ
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Solution:
$$\eqalign{ & \frac{{\sec \theta \left( {\sin \theta - 2{{\sin }^3}\theta } \right)}}{{2{{\cos }^3}\theta - \cos \theta }} \cr & \frac{{\sec \theta .\sin \theta \left( {1 - 2{{\sin }^2}\theta } \right)}}{{\cos \theta \left( {2{{\cos }^2}\theta - 1} \right)}} \cr & \frac{{\sec \theta .\sin \theta \times \cos 2\theta }}{{\cos \theta \times \cos 2\theta }} \cr & \sec \theta .\tan \theta \cr} $$
77.
The value of $$\left[ {\frac{{{\text{co}}{{\text{s}}^2}{\text{A}}\left( {{\text{sin A}} + {\text{cos A}}} \right)}}{{{\text{cose}}{{\text{c}}^2}{\text{A}}\left( {{\text{sin A}} - {\text{cos A}}} \right)}} + \frac{{{\text{si}}{{\text{n}}^2}{\text{A}}\left( {{\text{sin A}} - {\text{cos A}}} \right)}}{{{\text{se}}{{\text{c}}^2}{\text{A}}\left( {{\text{sin A}} + {\text{cos A}}} \right)}}} \right]$$ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ $$\left( {{\text{se}}{{\text{c}}^2}{\text{ A}} - {\text{cose}}{{\text{c}}^2}{\text{ A}}} \right) = ?$$
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Solution:
$$\left[ {\frac{{{\text{co}}{{\text{s}}^2}{\text{A}}{\text{.si}}{{\text{n}}^2}{\text{A}}\left( {{\text{sin A}} + {\text{cos A}}} \right)}}{{\left( {{\text{sin A}} - {\text{cos A}}} \right)}} + \frac{{{\text{si}}{{\text{n}}^2}{\text{A}}{\text{.co}}{{\text{s}}^2}{\text{A}}\left( {{\text{sin A}} - {\text{cos A}}} \right)}}{{\left( {{\text{sin A}} + {\text{cos A}}} \right)}}} \right]$$ $$\left[ {\frac{1}{{{\text{co}}{{\text{s}}^2}{\text{A}}}} - \frac{1}{{{\text{si}}{{\text{n}}^2}{\text{A}}}}} \right]$$ $$ \Rightarrow \left[ {\frac{{{{\left( {{\text{sin A}} + {\text{cos A}}} \right)}^2} + {{\left( {{\text{sin A}} - {\text{cos A}}} \right)}^2}}}{{\left( {{\text{sin A}} - {\text{cos A}}} \right)\left( {{\text{sin A}} + {\text{cos A}}} \right)}}} \right]$$ $$\left( {{\text{si}}{{\text{n}}^2}{\text{ A}} - {\text{co}}{{\text{s}}^2}{\text{ A}}} \right)$$ $$\eqalign{ & \Rightarrow 2\left( {{\text{si}}{{\text{n}}^2}{\text{ A}} + {\text{co}}{{\text{s}}^2}{\text{ A}}} \right) \cr & \Rightarrow 2 \cr} $$
78.
If sec15ÃÂÃÂÃÂø = cosec15ÃÂÃÂÃÂø (0ÃÂÃÂÃÂð ÃÂÃÂÃÂø 10ÃÂÃÂÃÂð) then the value of ÃÂÃÂÃÂø is?
(A) 9°
(B) 5°
(C) 8°
(D) 3°
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Solution:
$$\eqalign{ & \sec 15\theta = \operatorname{cosec} 15\theta \cr & \Rightarrow \frac{1}{{\cos {{15} }\theta }} = \frac{1}{{\sin 15\theta }} \cr & \Rightarrow \frac{{\sin 15\theta }}{{{\text{cos15}}\theta }} = 1 \cr & \Rightarrow {\text{tan15}}\theta = 1 \cr & \Rightarrow {\text{tan15}}\theta = {\text{tan}}{45^ \circ } \cr & \Rightarrow 15\theta = {45^ \circ } \cr & \Rightarrow \theta = \frac{{45}}{{15}} \cr & \Rightarrow \theta = {3^ \circ } \cr} $$
79.
The value of $$\frac{{3\left( {{{\cot }^2}{{47}^ \circ } - {{\sec }^2}{{43}^ \circ }} \right) - 2\left( {{{\tan }^2}{{23}^ \circ } - {\text{cose}}{{\text{c}}^2}{{67}^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}\left( {{{68}^ \circ } + \theta } \right) - \tan \left( {\theta + {{61}^ \circ }} \right) - {{\tan }^2}\left( {{{22}^ \circ } - \theta } \right) + \cot \left( {{{29}^ \circ } - \theta } \right)}}\,{\text{is:}}$$
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Solution:
$$\eqalign{ & \frac{{3\left( {{{\cot }^2}{{47}^ \circ } - {{\sec }^2}{{43}^ \circ }} \right) - 2\left( {{{\tan }^2}{{23}^ \circ } - {\text{cose}}{{\text{c}}^2}{{67}^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}\left( {{{68}^ \circ } + \theta } \right) - \tan \left( {\theta + {{61}^ \circ }} \right) - {{\tan }^2}\left( {{{22}^ \circ } - \theta } \right) + \cot \left( {{{29}^ \circ } - \theta } \right)}} \cr & = \frac{{3\left( {{{\tan }^2}{{43}^ \circ } - {{\sec }^2}{{43}^ \circ }} \right) - 2\left( {{{\tan }^2}{{23}^ \circ } - {{\sec }^2}{{23}^ \circ }} \right)}}{{{\text{cose}}{{\text{c}}^2}\left( {{{68}^ \circ } + \theta } \right) - \tan \left( {\theta + {{61}^ \circ }} \right) - {{\cot }^2}\left( {{{68}^ \circ } - \theta } \right) + \tan \left( {{{61}^ \circ } - \theta } \right)}} \cr & = \frac{{3 \times \left( { - 1} \right) - 2 \times \left( { - 1} \right)}}{1} \cr & = \frac{{ - 3 + 2}}{1} \cr & = - 1 \cr} $$
80.
If tanÃÂÃÂÃÂø - cotÃÂÃÂÃÂø = 0 find the value of sinÃÂÃÂÃÂø + cosÃÂÃÂÃÂø ?
(A) 0
(B) 1
(C) $$\sqrt 2 $$
(D) 2
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Solution:
$$\eqalign{ & {\text{tan}}\theta - \cot \theta = 0 \cr & {\bf{Shortcut\,\, method:}} \cr & {\text{Put }}\theta = {45^ \circ } \cr & {\text{tan }}{45^ \circ } - \cot {45^ \circ } = 0 \cr & 1 - 1 = 0 \cr & 0 - 0({\text{matched}}) \cr & So,\theta = {45^ \circ } \cr & \Rightarrow \sin \theta + \cos \theta \cr & \Rightarrow \sin {45^ \circ } + \cos {45^ \circ } \cr & \Rightarrow \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }} \cr & \Rightarrow \sqrt 2 \cr} $$