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worksheet class 1 maths

worksheet class 1 maths

If ${a_1},{a_2},{a_3}, \ldots ,{a_n}$ is an arithmetic progression with common difference
$d$, then evaluate the following Exemplarpression.
$\tan \left[ {{{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_1}{a_2}}}} \right) + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_2}{a_3}}}} \right) + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_3}{a_4}}}} \right)} \right.$ $\left. { + \ldots + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_{n – 1}}{a_n}}}} \right)} \right]$

We have, ${a_1} = a,{a_2} = a + d,{a_3} = a + 2d$and $d = {a_2} – {a_1} = {a_3}…

Show that $\tan \left( {\frac{1}{2}{{\sin }^{ – 1}}\frac{3}{4}} \right) = \frac{{4 – \sqrt 7 }}{3}$ and justify why the other value $\frac{{4 + \sqrt 7 }}{3}$ is ignored?
[NCERT,Exemplar.2.3,Q.18,Page.37]

Find the value of $4{\tan ^{ – 1}}\frac{1}{5} – {\tan ^{ – 1}}\frac{1}{{239}}$.
[NCERT,Exemplar.2.3,Q.17,Page.37]

Prove that ${\tan ^{ – 1}}\frac{1}{4} + {\tan ^{ – 1}}\frac{2}{9} = {\sin ^{ – 1}}\frac{1}{{\sqrt 5 }}$.
[NCERT,Exemplar.2.3,Q.16,Page.36]

Show that ${\sin ^{ – 1}}\frac{5}{{13}} + {\cos ^{ – 1}}\frac{3}{5} = {\tan ^{ – 1}}\frac{{63}}{{16}}$.
[NCERT,Exemplar.2.3,Q.15,Page.36]

Prove that ${\sin ^{ – 1}}\frac{8}{{17}} + {\sin ^{ – 1}}\frac{3}{5} = {\sin ^{ – 1}}\frac{{77}}{{85}}$.
[NCERT,Exemplar.2.3,Q.14,Page.36]

Find the simplified form of
${\cos ^{ – 1}}\left( {\frac{3}{5}\cos x + \frac{4}{5}\sin x} \right)$, where $x \in \left[ {\frac{{ – 3\pi }}{4},\frac{\pi }{4}} \right]$.
[NCERT,Exemplar.2.3,Q.13,Page.36]

Prove that ${\tan ^{ – 1}}\left( {\frac{{\sqrt {1 + {x^2}} + \sqrt {1 – {x^2}} }}{{\sqrt {1 + {x^2}} – \sqrt {1 – {x^2}} }}} \right) = \frac{\pi }{4} + \frac{1}{2}{\cos ^{ – 1}}{x^2}$.
[NCERT,Exemplar.2.3,Q.12,Page.36]

Solve the equation $\cos \left( {{{\tan }^{ – 1}}x} \right) = \sin \left( {{{\cot }^{ – 1}}\frac{3}{4}} \right)$.
[NCERT,Exemplar.2.3,Q.11,Page.36]

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