Question

Express all the trigonometric ratios of an angle A in terms of TanA.

​

Answer 1

Answer:

We are aware of the combinations of trigonometric ratios by expressing one ration into other ratio, such that

\sin x=\frac{1}{\cosec x}sinx=

cosecx

1

\cos x=\frac{1}{\sec x}cosx=

secx

1

\tan x=\frac{1}{\cot x}tanx=

cotx

1

Trigonometric identities:

\sin x^{2}+\cos x^{2}=1sinx

2

+cosx

2

=1

\sec x^{2}-\tan x^{2}=1secx

2

−tanx

2

=1

\cosec x^{2}-\cot x^{2}=1cosecx

2

−cotx

2

=1

By using the above identities, we can represent one ratio into another format.

\sin x=\frac{1}{\cosec x}sinx=

cosecx

1

=\sqrt{\frac{1}{\cosec x^{2}}}=

cosecx

2

1

=\sqrt{\frac{1}{1+\cot x^{2}}}=

1+cotx

2

1

=\sqrt{\frac{\tan x^{2}}{1+\tan x^{2}}}=

1+tanx

2

tanx

2

\cos x=\frac{1}{\sec x}cosx=

secx

1

=\sqrt{\frac{1}{\sec x^{2}}}=

secx

2

1

=\sqrt{\left(\frac{1}{1+\tan x^{2}}\right)}=

(

1+tanx

2

1

)

\cosec x=\frac{1}{\sin x}cosecx=

sinx

1

By using above sine representation the value of \frac{1}{\sin x}

sinx

1

is

=\frac{1}{\sqrt{\left(\frac{\tan x^{2}}{1+\tan x^{2}}\right)}}=

(

1+tanx

2

tanx

2

)

1

c\sec x=\sqrt{\sec x^{2}}csecx=

secx

2

=\sqrt{1+\tan x^{2}}=

1+tanx

2

\cot x=\frac{1}{\tan x}cotx=

tanx

1