Find the value of  \(sin \left( { 45 }^{ o }+\theta \right) -cos\left( { 45 }^{ o }-\theta \right) .\)

1

(a)

\(sin\quad \left( { 45 }^{ o }+\theta \right) -cos\left( { 45 }^{ o }-\theta \right)\)
\(=(sin{ \quad 45 }^{ o }\quad cos\quad \theta +cos\quad { 45 }^{ o }\quad sin\quad \theta )-(cos\quad { 45 }^{ o }\quad cos\quad \theta +sin\quad { 45 }^{ o }\quad sin\theta )\)
\(=\quad 0\)

If \(tan \alpha =\frac { m }{ m+1 } \) and \(tan \beta =\frac { 1 }{ 2m+1 } ,\) then find the value of  \(\alpha +\beta \)

1

(a)

\(tan\quad \left( \alpha +\beta \right) =\frac { tan\quad \alpha \quad +\quad tan\quad \beta }{ 1-\quad tan\quad \alpha \quad tan\quad \beta } \)
\(=\frac { \frac { m }{ m+1 } +\frac { 1 }{ 2m+1 } }{ 1-\frac { m }{ m+1 } \times \frac { 1 }{ 2m+1 } } \)
\(\Rightarrow \quad tan\quad \left( \alpha +\beta \right) =1\therefore \alpha +\beta =\frac { \pi }{ 4 } \)

Find the value of tan 3A - tan 2A - tan A.

1

(a)

tan 3A = tan (2A + A)
\(\Rightarrow \quad tan\quad 3A=\frac { tan\quad 2A\quad +\quad tan\quad A }{ 1-tan\quad 2A\quad tan\quad A } \)

If \(sin\theta +cosec\theta =2\), then find the value of \({ sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta .\)

1

(a)

\((sin\theta +cosec\theta )^{ 2 }=4\quad \Rightarrow { sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta +2=4\)
\(\Rightarrow { sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta \quad =2\)

Which of the following are correct?
(i) sin1⁰>sin1
(ii) sin1⁰
(iii) sin1⁰=sin1
(iv) \(sin{ 1 }^{ 0 }=\frac { \pi }{ 180 } sin1\)

1

(a)

(ii) sinl° < sin1