Express tan theta in terms of cosec theta
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tan theta = 1\ cot theta. tan theta = √1/cot² theta tan theta = √1/cosec² theta - 1 (W.K.T cosec² theta - cot² = 1, cot² theta = cosec² theta - 1) tan theta = √1/cosec² theta - 1
To express tan theta in terms of cosec theta, we need to use the trigonometric identity that relates these two functions.
The trigonometric identity is:
tan theta = 1 / cosec theta
Let's break down how to obtain this identity.
1. The reciprocal of sine is cosec (cosecant). Hence, we have:
cosec theta = 1 / sin theta
2. Now, we need to express tan theta in terms of sin theta. The relationship between tangent and sine is:
tan theta = sin theta / cos theta
3. Next, replace sin theta with 1 / cosec theta, which we derived in step 1:
tan theta = (1 / cosec theta) / cos theta
4. Simplify the right-hand side of the equation by multiplying numerator and denominator by cos theta:
tan theta = (1 / cosec theta) * (1 / cos theta)
5. Use the property of multiplying reciprocals:
tan theta = 1 / (cosec theta * cos theta)
6. Finally, note that cosec theta * cos theta is equal to 1. This is a trigonometric identity:
tan theta = 1
Therefore, tan theta expressed in terms of cosec theta is simply 1.
Very bad who solved this he is sense less fellow what is the question I asked and what is the answer given
csc = 1/sin
so
sin = 1/ csc
sin^2 = 1/csc^2
cos^2 = 1 - sin^2 = 1 - 1/csc^2
so
tan^2 = sin^2/cos^2 = (1/csc^2)/[1-1/csc^2]
tan^2 = 1/[csc^2 - 1]
tan = sqrt (1/[csc^2 - 1] )
Or, more directly,
tan = 1/cot = 1/√(cot^2) = 1/√(csc^2-1)